RADIATION INDUCED NUCLEATION
IN
WATER AND ORGANIC LIQUIDS
by
Nicholas P. Oberle
B.S.M.E., Polytechnio Institute of Brooklyn
(1970)
Submitted in Partial Fulfillment of
the Requirements for the Degree of
Master of Science
at the
Massachusetts Institute of Technology
March 1972
Signature of Author
Certified by
_
Thesis Supervisor, Date
Accepted by
Chairman, Departmental Committee
on Graduate Students
Archives
,IUN 13 1972
2
Radiation Induced Nucleation
in
Water and Organic Liquids
by
Nicholas P. Oberle
Jubmitted to the Department of Nuclear Engineering
on March 24, 1972 in partial fulfillment of the requirements
for the degree of Master of Science.
Abstract
Experimental data for the threshold superheat
was obtained for the case of fission fragments in water
and in propylene glycol. Analytical results, using
Bell's (5) energy balance method, for the threshold
superheat values were obtained for both fission fragments
and fast neutrons in water, propylene glycol, benzene,
acetone, and nitromethane.
Experimental and analytical results were compared
with data obtained by other investigators.
(3),(4),(5).
A detailed discussion of the liquid suspension
method used in this work is given, along with a criteria
for selecting a test fluid. and its supporting fluids.
Physical properties vs. temperature plots for a
number of possible test fluids are also given.
Thesis Supervisor
Title
Neil E. Todreas
Associate Professor
Acknowledgements
I wish to express my sincere gratitude to my
thesis supervisor, Professor Neil E. Todreas, for his
advice and assistance throughout this study. I would
also like to express my appreciation to Dr. Norman C.
Rasmussen for being kind enough to be my thesis reader.
Special thanks are also due to Dr.
C.R.Bell for
his helpful advice. I would like to take this opportunity
to thank all members of the M.I.T. reactor machine shop
and. radiation protection office for their technical
advice and assistance.
Last but not least, I would like to thank all those
friends in the Nuclear Engineering Department and in
the Chemistry Department who had. in one way or another
assisted me in the present work.
Table of Contents
Abstract
Acknowledgements
List of Figures
List of Tables
Chapter 1.
Introduction
1.1
Radiation Induced Nucleation
10
1.2
Experimental Methods
12
1.2a
Expansion Method.
12
1.2b
Constant Pressure Method
13
1.2c
Advantages and Disadvantages
13
Objectives of Present Work
14
1.3
Chapter 2.
Theory
2.1
General Introduction
16
2.2
The Electrostatic Theory
16
2.3
Thermal Theory
19
2.4
Minimum Energy for Bubble Formation
20
2.5
Energy Balance Method of Bell
28
2.6
Fission Fragments and Fast Neutrons
41
Chapter 3.
Experimental Apparatus and Procedures
3.1
Apparatus Modifications
43
3.2
Experimental Procedure
51
3.3
Experimental Difficulties
55
3.4
Indication of Nucleation
57
3.5
Fission Fragments in a Test Liquid
58
Chapter 4.
Fluid Selection
4.1
Background Information
60
4.2
Test Liquid Properties
60
4.3
Compatibility Requirements
62
4.4
Difficulties in Fluid Selection
63
4.5
Discussion of Surface Tension
71
Chapter 5.
Results
5.1
Analytical Results
76
5.2
Experimental Results
87
5.3
SenSitivity Study of Superheat Data
90
5.4
Comparison with other Results
96
Chapter 6.
Conclusions and Recommendations
6.1
Conclusions
103
6.2
Rayleigh's Criteria
105
6.3
Recommendations
108
Appendix A.
Physical Properties
A.1
Introduction
112
A.2
Physical Properties of Acetone
114
A.3
Physical Properties of Diethyl Ether
120
A.4
Physical Properties of Propylene Glycol 121
A.5
Physical Properties of Nitromethane
127
A.6
Discussion of Vapor Density Results
128
A.7
Physical Properties of Methyl Alcohol
130
A.8
Physical Properties of Ethyl Alcohol
136
Appendix B.
Sample Calculation
137
Appendix C.
Cpmputer Programs
147
Appendix D.
References
168
___________IIIII_1_11____1~-
-~·L_~II-·-·---LI·LIP·
List of Figures
Page
Figure
2.1
Thermodynamic systems for bubble formation.
22
2.2
Typical energy loss for a heavy charged
particle interacting with matter.
29
2.5
Pg routine flowchart
38
2.4
Threshold superheat flowchart
40
3.1
Diagram of Experimental
44
3.2
Diagram of Convection Generator
47
3.3
Light Source
48
3.4
Mirror system
50
3.5
Source Holder and Shielding
52
3.6
Pressure Gauge calibration Curve
53
4.1
Three-Liquid configuration
62
4.2
Spreading Effect of Test Liquid
68
5.
Threshold Superheat vs Pressure for
5.1
Fast neutrons in water
77
5.2
Fission fragments in water
78
5.3
Fast neutrons in Propylene Glycol
79
5.4
5 05
Fission Fragments in Propylene Glycol
Fast Neutrons in Benzene
80
81
5.6
Fission Fragments in Benzene
82
5.7
Fast Neutrons in Acetone
83
5.8
Fission Fragments in Acetone
84
5.9
Fast Neutrons in Nitromethane
85
5.10
Fission Fragments in Nitromethane
86
5.11
Fission Fragments in water with
experimental results
88
5.12
Fission
5.13
Fission Fragments in water with + 7%
uncertainty in surface tension.
93
5.14
Fission Fragments in water with + 20%
94
5.15
Fission fragments in water with a
95
combined uncertainty of +7% in
surface tension and +20.-in mean charge
5.16
Fission fragments in Propylene Glycol
with a +5% uncertainty in surface
tension
97
5.17
Fission frrgments in Propylene Glycol
with a +20% uncertainty in the
mean charge.
98
5.18
Fission Fragments in Propylene Glycol
with a combined uncertainty of +5% in
surface tension, and +20% in mean
charge.
99
5.19
Fission fragments in water-data from
Deitrich and Oberle.
100
5.20
Fast neutrons in Benzene , with
E1-Nagdy data.
101
5.21
Fast neutrons in Acetone, with
E1-Nagdy data.
102
Fragments in Propylene Glycol
with exrerimental results.
uncertainty in mean charge.
91
6.1
Liquid-Vapor jet configurations.
106
6.2
Proposed experimental apparatus.
110
A.1
Vapor Pressure vs. Temperature plot for
Acetone and. Diethyl Ether.
115
A.2
Vapor Density vs. Temperature plot for
Acetone and. Diethyl Ether.
116
A.3
Liquid Density vs. Temperature plot for
Acetone and Diethyl Ether.
117
A.4
Surface Tension vs. Temperature plot for
Acetone and Diethyl Ether.
118
A.5
Heat of Vaporization vs. Temperature plot
for Acetone and Diethyl Ether.
119
A.6
Vapor Pressure vs. Temperature for
Propylene Glycol and Nitromethane
122
A.7
Liquid Density vs. Temperature for
Propylene Glycol and Nitromethane
123
A.8
Vapor Density vs. Temperature for
Propylene Glycol and Nitromethane
124
A.9
Surface Tension vs. Temperature for
Propylene Glycol and Nitromethane
125
A.10
Heat of Vaporization vs. Temperature for
Propylene Glycol and Nitromethane
126
A.11
Vapor Pressure vs. Temperature for
Methyl and Ethyl alcohol
131
A.12
Liquid Density vs. Temperature for
Methyl and Ethyl alcohol
132
A.13
Vapor Density vs. Temperature for
153
Methyl and Ethyl alcohol
A.14
Surface Tension vs. Temperature for
Methyl and Ethyl alcohol
134
A.15
Heat of Vaporization vsa, Temperature for
-ethyland Ethyl alcohol
135
List of Tables
Table
2.1
Affect of additional surface tension
term
27
4.1
Miscibility chart for benzene and acetone
65
4.2
Liquids immiscible with benzene
66
4.3
Liquids immiscible with acetone
66
4.4
Partial list of three-liquid tests
69
5.1
Results from Tso and Oberle for fission
fragments in water
87
5.2
Results from Tso and Oberle for fast
neutrons in water
89
10
Chapter 1
Introduction
1.1
Radiation Induced Nucleation
The term radiation-induced nucleation refers to the
phenomena of nucleation (the initial formation of bubbles)
of vapor or gas bubbles in liquids due to localized energy
deposition or localized ionization along the track of an
energized particle.
Homogeneous nucleation is normally not considered an
important part of the boiling process in engineering
components, since nucleation due to the presence of
surface cavities, suspended particles, or non-condensible
gas bubbles is the dominant mechanism at small liquid
superheats. Pure homogeneous nucleation only becomes
important at high superheats.
This phenomena of radiation-induced nucleation of
bubbles in a superheated liquid was first reported by
D.A. Glaser (1), who was searching for an instrument
similar to a cloud-chamber-like detector, which would
aid in the study of high energy nuclear events. The cloud
chamber utilizes the instability of supercooled vapors
against droplet formation, so it seemed promising to
attempt to devise an instrument which could take advantage
of superheated liquids against bubble formation. Glaser
11
made an experimental test with superheated diethyl ether.
In the absence of any radioactive source a superheated
state was able to be maintained for a few seconds. But
when a radioactive source was brought near the superheated
liquid, it immediately erupted into vapor.
This series of experiments showed that bubble chambers
could be a practical device for the study of high energy
nuclear events, and led to the highly sophisticated bubble
chamber technology which exists today.
Since the existence of the radiation-induced nucleation
phenomena was thus demonstrated, the question soon arose
of whether or not this would have any affect on nuclear
reactor coolants or moderators. Recently, it was desired
to know whether this phenomena would have any affect on
the sodium void problem in sodium cooled fast reactors.
Liquid sodium can attain very high superheats on engineering
surfaces. Nucleation of this highly superheated sodium
will lead to violent flashing, resulting in the sudden
ejection of sodium from one or more coolant channels.
The rate of sodium ejection is related to the amount
of superheat attained. Since a reactor is an intense
source of radiation, the possibility that this radiation,
through radiation-induced nucleation, might place an
upper limit on the superheat attainable in reactor
coolants was a factor worth studying.
12
1.2
Experimental Methods
There are presently two experimental methods
employed to study the radiation-induced nucleation
phenomena. The major difference between these two methods
is in the manner in which the superheated state of the
liquid is attained.
The first, and up to now the most widely used method,
is the so-called expansion method. The second is a
constant pressure type method. Both of these methods will
be reviewed, emphasizing both their advantages and
disadvantages.
1.2a
Expansion Method
The expansion method is similar to the bubble chamber
method of operation. The test liquid is heated to a
temperature considerably above its normal boiling point,
but is kept at a pressure which still maintains it as a
stable liquid.(In other words, the test liquid is kept
under a pressure higher than the saturated vapor pressure
corresponding to the temperature to which the liquid is
raised.) The expansion process then occurs. (Expansion is
the sudden lowering of the pressure to a predetermined
pressure at which the test liquid is superheated.) The
pressure is thus reduced from an initial value, at which
the liquid is subcooled, to a lower value,where the test
liquid is superheated. This method has been used by
Becker(2), Deitrich (3), and El-Nagdy (4).
1.2b
Constant Pressure Method
This method, devised by Bell (5),
attains the
superheated state of the test fluid by maintaining a
constant pressure and gradually heating the test liquid.
Nucleation from the surface of the test liquid does not
occur since the test liquid is not in contact with a solid
surface. This is accomplished by suspending the test
liquid in the form of a bubble (approximately one-half
inch in diameter ) between two other liquids, a supporting
fluid and a cover fluid. A detailed description of this
experimental technique will be given in a later section
of this work.
1.2c
Advantages and Disadvantages
A major disadvantages of the expansion method is
the short duration (a few seconds ) of the superheated
state of the test liquid. Any effect of radiation on the
liquid must therefore be observable during this short
period of superheat..
If a reactor ( as in Deitrich (3) ) or an accelerator
( as in El-Nagdy (4) ) is used as the source of radiation,
then a further complication, that of synchronizing the
radiation burst with the expansion in the test chamber,
is encountered.
Another disadvantage of the expansion method is
that of surface cavities. These will cause boiling on
surfaces of the chamber well before the radiation induced nucleation phenomena can appear. ( This problem
14
has been solved to some degree by El-Nagdy (4), by cooling
the test chamber walls just prior to and during the
expansion process.)
The constant pressure heating method can maintain a
high degree of superheat for an indefinite period of
time in the absence of radiation since the technique of
suspending the test bubble eliminates surface effects.
The one main disadvantage of this method is that the
requirements of this three-liquid configuration are such
that it is very difficult to find compatible supporting
and cover fluids for many test fluids of interest. The
expansion method on the other hand has no such liquid
compatibility requirement, and can be used for any liquid
desired. A detailed discussion of these compatibility
requirements will be given later.
1.3
Objectives of Present Work
The constant pressure heating method was begun at
M.I.T.
by C.R. Bell (5) who formulated a theory and
constructed the experimental apparatus. Some preliminary
experimental work with water as the test fluid was also
done by Bell (5).
This work was continued by Tso (6), who
extended the experimental work with water, and performed
preliminary analytical work with benzene.
The present work included experimental work with water
and propylene glycol, and analytical work (utilizing
Bell's method ) with water, propylene glycol, benzene,
acetone, and nitromethane. Modifications were also made
to the experimental apparatus.
16
Chapter 2
Theory
2.1
General Introduction
Two theories have been proposed to explain the
formation of bubbles by radiation. The first theory, the
"Electrostatic theory", was proposed by Glaser (7), and
the second theory, the "thermal spike" theory, was
proposed by Seitz (8).
Actually there are only two ways in which energy can
be extracted from ions to produce bubbles: either the
energy of the electric field due to the charges of the
ions is converted into the work of formation of bubbles,
or the recombination energy of the ions is given off as
heat, which produces a so called "thermal spike", and
finally bubbles.
The thermal spike theory has generally been accepted
as the correct theory, but a short discussion on the
electrostatic theory will be-presented, since it also
did a reasonable job of predicting operating conditions
for a number of bubble chambers.
This thesis will use the theoretical approach of
Bell (5), which is based on the thermal spike theory.
2.2
The Electrostatic Theory
In order to predict the operating conditions for
17
a bubble chamber, a theory was needed to quantitatively
describe the conditions under which ionizing events
could nucleate bubbles in a superheated liquid. Glaser(7)
then developed his electrostatic theory.
The electrostatic theory was a microscopic model
of an ionization-triggered bubble nucleation mechanism.
It was assumed that an ionizing particle produces clusters
of electric charges of like sign within a liquid, and
that the mutual electrostatic repulsion of these charges
could then fracture a superheated liquid to produce
bubble nuclei large enough to grow to visible size, using
the thermal energy from the liquid. An approximate
treatment of the effect of electrostatic forces on bubble
nucleation was made, which assumed a number "N" (it was
known that a number of charges were required for this
mechanism to work) of electronic charges to be distributed
uniformly over the surface of a potential bubble nucleus.
This treatment formulated an "electrostatic pressure
effect", which predicted that a bubble carrying "N" like
charges would grow if the pressure on the liquid were
less than the saturated vapor pressure at the ambient
temperature by a certain amount. This difference,
AP
and
=
P,(T) - P
, where:
P is the pressure on the liquid
P,(T) is the saturated vapor pressure over a flat
liquid -vapor interface,
> -3
is
(p
given by:
)~i·(TI
-3
4
Cr(2,]
(2. )
18
where:
e
= electronic charge
0~(T)= surface tension of the liquid at temperature T
(T) = dielectric constant of the liquid at temperature T
With N taken as 6, this formula had fairly good
success in predicting the operating conditions of a
number of liquids, such as hydrogen, deuterium, and
helium.
Even though the theory met with some initial success,
serious objections arose. One of these considered the
fact that in order to rupture the superheated liquid,
the N charges must be deposited within a sphere of radius
"r",
where r is given bys
PC(r)p
-YTJL
For typical operating conditions for certain liquids,
such as some organic liquids, this gives a radius of the
order of 10- 6 cm.. This requires that an ionizing particle
must leave a cluster of six charges, of the same sign, in
this small volume. It was obvious that this does not
happen often enough in primary ionizations to account
for the observed number of bubbles.
A contradiction with the theory was also found when
alpha particles were used as the ionizing agent. To agree
19
with this theory, the alpha particle was required to
deposit 900 charges, of the same sign, in a region
2x10-6cm. in diameter. This is greater than the maximum
ionization attained by an alpha particle during the slowing
down process, even granting the possibility of the
required charge separation. The total energy loss of the
alpha particle, however, was sufficient to explain the
observed effects, so another model for the microscopic
mechanism was sought. ( For a more detailed description
of the electrostatic theory and its disadvantages,
references (4), (9), and (10) are recommended. )
2.3
Thermal Theory
The thermal theory suggests that local heating
by ions and S -rays (knocked-on electrons) could produce
bubbles along the track of ionized particles.
It was assumed that the primary mechanism for the
deposition of this local thermal energy occurred through
the ý -rays formed by the primary ionizing particle. These
knock-on electrons would lose energy by collisions with
molecules.
The energy of the
f-ray is thus rapidly
converted to kinetic energy of the molecules of the
medium. The primary ionizing particle thus produces
along its path a series of "thermal spikes", which then
act as nuclei for bubble formation.
Seitz (8) suggested that this thermal energy
could also be obtained from the heat energy released
20
in ion recombination at the end of
j
-rays,
rather
than just coming from molecular excitation.
The basic assumption of this thermal spike model
is
that the minimum energy required to form a bubble
of critical size is provided from the thermal spike
created in a small initial volume by the incident particle.
This minimum energy ( Em ) required for bubble formation
will be given in the next section.
2.4
Minimum Energy for Bubble Formation
The minimum energy (Er) required for the formation
of a critical nucleus is composed of a number of terms.
Norman and Spiegler (11)
give Em as:
3
where:
R
Rc = critical bubble radius
= density
dH = enthalpy change per unit mass
-
= surface tension
7
= temperature
A
R
r
F-
velocity of the bubble wall
viscous losses
and the subscripts
liquid states.
v
and
1
refer to the vapor and
(2.3)
21
The first term on the RHS is
the energy required
to heat and vaporize the mass of liquid involved and
expand it against the external pressure. The second term
is
the energy required to form a surface in the liquid.
The third term is
the kinetic energy given off to the
liquid by the motion of the vapor wall. An expression
for R is
given (11)
*i
where
below
YD
D is
the thermal diffusivity.
The viscous loss term,
(2), (12)
F
, has been shown ( (11),
) to be negligible compared to the other terms.
Deitrich (12) also uses equation (2.3) as the Em
required for bubble formation.
Becker (2) uses the following equation for E
Em-
3
- 'r/?c
1
+
7RR
-
Rc
:
7)
(2.5)C
(2.5)
Seitz (8), El-Nagdy (4), and Bell (5) use the
Em equation given below;
E, -
Z7r(RC-1
e
RC%
)
Bell ( (5) pg. 257 ) has also derived equation (2.6)
from a purely thermodynamic viewpoint, which will be
considered below.
(2.6)
22
Two systems are shown below. System I is liquid
only, while system II contains a vapor bubble in the
liquid. (See FIG.2.1 below ) The energy required to go
from system I to system II will be found.
II
FIG. 2.1
The system parameters are defined as follows:
m = mass of the fluid
P = pressure
T = temperature
V = volume , v = Specific volume
The subscripts v and 1 again refer to the vapor and
liquid states.
23
From the first law of thermodynamics
(Jc --=- Qn:
-
W-(2.7)
Ux = internal energy of system x
where
Qx
=
energy added to system x
Wx = work done by system x
The total energy terms are now converted to specific
terms,
("It h-y)
~my -e a -=
. + myV~
--
(2.8)
The work done by the system (WII ) is composed of two
parts: the work done to create the void, and the work
done to form the surface. WII is
Wn, r where
P-
VV
)
given by
-AE
(2.9)
0
a surface tension of embryo
A
= surface area of the vapor embryo
Substituting equation (2.9) into (2.8) and rearranging
terms,
,e'v
,rv
24
Now Pvv, is added to both sides of equation (2.10):
(QV t- P r
SP
V
(2.11)
From the definition of enthalpy:
(2.12)
h = u + Pv
Substituting this into equation (2.11)
,
(h,--
Q
Q
Now, assuming (5)
t,
-
s
P)
-Av1
(2.13)
that the change in enthalpy (hv-hl)
is approximately equal to the heat of vaporization:
hfg • h
hI
(2. 14)
Also, a number of substitutions can be made:
rnv-
r
V
(2. 15)
(2. 16)
/
(2.17)
where
a density
-y
=
bubble radius.
For a critical embryo the equilibrium condition gives;
V
IRC
(2.18)
25
Utilizing the above substitutions, equation
(2.11) is
reduced to:
3
3
ii 7
~
(2.19)
where QII is the energy which must be added to the
system in order to create the critical embryo.( In
other words, QII = Em )
Equation (2.19) is exactly the same equation as
the Em equation employed by Seitz, El-Nagdy, and later
by Bell. (Equation (2.6) ) This is the equation which
was used in the present work.
It was noticed that some investigators had included
an additional surface tension term in their Em equation (11).
(3),
(2). This term was of the form
7-dT
Seitz(8), El-Nagdy(4), and Bell(5) did not include
this term. It was thus decided to see whether or not this
additional term was significant. If this term were
significant, then the original Em equation
'oul be
hae(2.20)
would be changed to
'r
C
/_ df_
(2.21)
26
Calculations were made for each of the liquids studied
to see whether or not this additional term would greatly
affect the Em equation.
The results of these calculations (TABLE 2.1)
show that the
/-
term is negligible for the liquids
studied (water, propylene glycol, nitromethane, benzene,
acetone).
The 9- term in itself is only a small fraction
of the Em equation. The addition of the /--- term does
dT-
not significantly change this fraction.
This is true for the liquids studied in this work.
But it does not necessarily hold for all liquids. In the
case of liquid hydrogen and deuterium ( used to a great
extent in bubble chambers ),
this additional term is seen
to significantly increase the contribution of the surface
tension terms.
Therefore, before this additional term can be
neglected, it must first be determined to what degree
this term affects the Em equation.
For the liquids studied in this work,
this term
was neglected, since it was proven to be negligible.
AFFECT
ýA--
v
h
OF ADDITIONAL SURFACE TENSION TERM
llbf
7
r
4f
__
__
pr,2
of A
r
"
ft
D,
%
afA
WATER
180.xio3
2.56xi03
1.4%
5.12x10
3
2.0%
PROPYLENE
GLYCOL
98.05x103
0.365xi0 3
0.3%
1.92x103
9%
NITRO-
METHANE
ACETONE
BENZENE
HYDROGEN
DEUTERIUM
193x103
0.88xi03
0.4%
3.35xio 3
1.7%
ACETONE
93.2x10 3
0.34x03
0.36%
0.88x10 3
0.95%
122.6x103
0.73xl 03
0.59%
1.89x103
1.5%
6.74%
13.05x103
6.09%
19.9oxi03
41.97xlo3
66.29x103
2.83x10 3
4.4xio03
TABLE 2.1
31.09%
30.02%
28
2.5
Energy Balance Method of Bell
In his work Bell(5) postulated that the energetic
particles involved in the nucleation process can be
thought of as producing long cylindrical regions of
energetic molecules along their path. Macroscopically,
this cylindrical region can be thought of as a vapor
cylinder. This vapor cylinder will have a diameter of
0
approximately 100 A ( (5) pg63). A vapor cylinder in
a liquid will tend to form into the more stable
configuration,
this case).
that of a sphere (or vapor bubble in
Since the total vapor track length is
several orders of magnitude greater than the critical
embryo diameter, it
is highly unlikely that the total
length of the vapor cylinder will reconfigure itself
into a single vapor sphere.
It
is more likely that
this long vapor cylinder will break up into smaller
fragments, and that these fragments will reconfigure
into individual embryos.
This process, as described by Bell(5),
is shown
schematically in FIG.2.2. Also shown is a typical
energy deposition curve for a heavy charged particle
interacting with matter.
In the first part of the track
the energy deposition to the electronic system of the
stopping medium is
the dominant energy transfer
mechanism, while in the later part of the track the
energy deposition to the medium through nuclear elastic
29
I
I
I
Radqiatton ra t
a
I
-I
-
I
VezP,r Cy/,nc/er
-
r
I
I
Sreak/<
up
/m
o
1l m6 ryoS
I
I
dE
ds
-
,
S,
s
S
sl
F'/G
2. Z
30
collisions will be dominant.
If a section of the radiation particle track length
is
taken (say s1 to s2,
particular embryo,
oras ) and assigned to one
then the energy available to this
embryo from the energetic radiation particle is
integral of the energy deposition rate,
the
dE/ds , over
the distance, A s , This energy available to the embryo
(or the energy lost by the particle) is given by
sA
iot -(s-)
-Sf)
C)]dS
where
J
(C) jS
(2.22)
is a function of the particle
energy at position s . But the energy deposition rate,
dE/ds, is a function of energy, and not position(13).
Therefore the integral in equation (2.22) cannot be
solved directly. An approximation has been made ((5),74)
for the average energy deposition rate:
-E(sA)
OR
EEsz)
d,
--
AZS
EJ,
ds
(2.23)
31
The minimum superheat, or the threshold condition,
for radiation-induced nucleation will be determined
by the largest value of A E along the particle track.
Assuming that the energy lost by the particle
is equal to the energy made available to the embryo,
and that the track breaks up in a uniform manner
(4 s = L = constant ,
(5)
pg.74 ),
value for AE will occur where
then the maximum
dE/ds is maximum,
which is at the beginning of the radiation track in
the stopping medium.
(Prom as
0,
to s = L, as shown
in FIG.2.2 ). E(0) would simply be the initial energy
of the energetic particle. If a value for A s = L is
assumed, then equation (2.23) would contain only one
unknown, E(L).
Bell (5)
assumed that this break up length is
given by
L = ar
r
where
is
(2.24)
the critical radius of the embryo
"a" is a dimensionless parameter
At this point it will only be mentioned that Bell ((5) 203)
postulated:
a = 6.07
. More will be said about this
dimensionless parameter in chapter 6.
The basic concept behind the energy balance
method is
that the energy available from the radiation
particle is equal to the energy required for bubble
nucleation. The energy available, Ea
,
is given by
32
equation (2.23). The energy required for bubble formation
was given by equation (2.6):
Em=
rr
(2.6)
There is another factor which must be added to
the above equations. This factor is
commonly referred
to as an "energy sink", and arises from the formation
of gas due to molecular dissociation in the stopping
medium. In the liquids studied in this work, hydrogen
is the gas produced by the dissociation process.
The total energy loss for all the hydrogen formed
in a track length L ( or ar
aE
=
Q and
where
) is
given by
QAEG(H 2 )10 6
(2.25)
NAV
G(H2 )
are constants, and aE
same AE as in equation (2.23).
is
the
This additional term
is usually subtracted from the available energy, Ea ,
term.
The final form of the energy balance equation
is thus:
Sx .3 9-
v
=10
Q,G
/
(2.26)
where the LHS is the minimum energy required for bubble
formation (Em) and the RHS is
Ea#
the total energy available,
33
When equation (2.26) is
satisfied, then the
conditions for radiation-induced nucleation have been
met.
Besides being an energy sink, the production of
the hydrogen gas also affects the dynamic equilibrium
of a critical embryo. The original mechanical equilibrium
condition for a critical embryo is given by:
p
-
2(2.27)
Taking the hydrogen gas into account, equation (2.27)
becomes:
P*
(2.28)
P, - PO
The effect of this hydrogen gas is to reduce the
vapor pressure requirement inside the critical nucleus
for a nucleus of a particular size, and thus reduce
the superheat required for that critical nucleus.
An equation for Pg (the gas pressure term) is ((5).
'-Y/"r
r 3
P
pg122)
(2.29)
cannot be calculated directly from equation (2.29),
since r* is a function of Pg ( in addition to being
a function of Pi), and A E is also a function of P,
since E(s2 ) depends on r
.
34
Before equation (2.29) can be solved, an expression
must be found for
dE/ds in equation (2.23).
A number
of energy deposition formulas have been suggested. The
one which is most widely used is
that given by Segre(13).
The
This has also been modified for fission fragments.
equation is given below:
/
y n on
electron
= charge
number
of an
molecules
of stopping medium per unit
volume ( molecular density)
(2.30)
The symbols used in equation (2.30) are defined below:
e tcharge
on an electron
h number of molecules of stopping medium per unit
volume ( molecular density)
x
o~= electron mass
V
Z
= velocity of the particle going through the
stopping medium
~= atomic number of the particle
Z;= atomic number of the ith atom in the stopping medium
= number of i
atoms per molecule
1
.= mean ionization potential of the ith component
I1
= mass of the particle (amu)
.
i=
=Mmass of the ith atom (amu)
Planck's constant divided by 2-ir
35
( Zi )eff
=
effective charge on the particle.
An approximate relationship for the above term is
given by Segre (13):
Z,
0.
= Z
(2.31)
e
= impact parameter beyond which energy loss by the
particle is effectively zero due to screening
of nuclei by atomic electrons for the ith atom.
An expression for the impact parameter is given by
Claxton (16) :
-
where
aH is
(2.32)
the radius of the first Bohr orbit for
the hydrogen atom.
Equation (2.30) must be evaluated for each
particular charged particle-stopping medium case. The
solution to equation (2.30) will have the general
form of
-•
where
-ir
(2.33)
A,B.C,D
:- is
E is
are constants
the density of the stopping medium
the energy of the radiation particle.
36
The first term in equation (2.33) represents
the energy deposition to the electronic system of the
stopping medium through ionization and excitation of
the atoms, and the second term represents the energy
deposition to the stopping medium through nuclear elastic
collisions between the energetic particles and the
nuclei of the stopping medium. In the cases considered
in this work, the second term is
negligible, and may
be neglected. ( This approximation is shown to be valid
by Bell (5) pg. 117, and by Segre (13) pg 166.)
Equqtion (2.33) can thus be reduced to
__
C-
,
13 C
(2.34)
ds
Equation (2.34) shows that the energy deposition
rate, dE/da , is a function of the particle energy, E.
This fact was mentioned earlier in connection with
equation (2.22).
Equation (2.34) is now substituted into
equation (2.23),
and the result takes the general form:
El(Cs,)s ) 6O)2nAz(S,)-(S 4 )kiB~a)
where
-R[Ers,)
J
Ca
B and C are constants
a
w Bell's parameter
= density of the stopping medium
r* = critical radius of embryo
rx
(2.35)
37
A substitution is now made for r*, using
equation (2.28):
r* =
2 -
(2.36)
PV+P -P1
It
is
now possible to calculate,
using an
iterative procedure, Pg,
The first step in this procedure is to guess a
value of Pg. r* is then calculated from equation (2.28)
( Pvo Pi, and
calculate
-are
known).
The next step is
E(s2 ) from equation (2.35). &E
to
can now
be found by recalling that
:E . E(sl) - E(s 2 )
E(si),
P
the initial particle energy, is always known.
is now calculated using equation (2.29). If this
calculated value of P
is
not equal to the original
Pg guess, then another value of Pg is assumed, and the
procedure is continued until the two values converge.
FIG. 2.3 gives a simplified flow chart of this calculational
procedure.
The convergence of this series will give correct
values of P
of P 1
' a
, r*
, and. 6E , for a given set of values
, and T1
Once Pg has been obtained, the superheat temperature
can be obtained. A complete program has been developed
by Bell((5), pg 122), which calculates the superheat
I
I
I
I
N
•
r"
s
CA LC LATE P
I
I
I
E(s~~~I
cALcuLATED
·t
I
I
IS
AE
________
__
is
1
CALCULATED
L
CALC UL ATE D
i
ROLU TI/ANE
FIG. 2.3
I
I...
threshold for various pressures and "a" values. It
is
an iterative procedure involving the system temperature,
T1 * A brief description of the program operation
will be given.
There are three independent variables: the
system pressure, Pi, the system temperature, T1 , and
the parameter "a" . P1 and "a" are considered independent
parameters. The first step of this procedure is to
select an initial value of T1, for a particular set
of P 1 and "a" values. This temperature value is then
fed into the Pg procedure mentioned earlier, ( the Pg
procedure can be considered a subprogram of the entire
calculational method for obtaining the threshold superheat.) The convergence of the P
routine gives the
corresponding values of Pg, r* , and AE , for the set
of chosen variables P1
, "a"
, and T1 . Using these
values of AE and r* , the LHS and RHS of equation (2.26),
are calculated. If
equation (2.26) is
a new value of T1 is
not satisfied,
chosen and the process is
repeated
until the energy balance criteria is met, When this
criteria is met, the T1 obtained is
the superheat
temperature corresponding to the threshold condition.
FIG. 2.4 gives a flowchart of the entire computational
process for obtaining the correct T1 value.
40
c,
I
II
|l
P
/RouOTI
2i
AT /5 CAL e L ATE-
oe
IN
zý7-
THRE/SHOLD S UPEY'HEA T
CALC ULA T/O/V
F-IG. 2.1'
41
2.6
Fission Fragments and Fast Neutrons
In the case of a fission fragment, the fragment
itself can directly produce the highly energetic
region in the medium which is necessary for radiationinduced nucleation. This is not true of a neutron.
The neutron interacts with the nucleus of an atom,
through elastic scattering, and transfers some of its
kinetic energy to the nucleus, thus producing a
charged primary knock-on atom ( PKOA ). This PKOA then
deposits its energy in a manner similar to the fission
framment.
The only change which this introduces into the
computer program, is in the initial energy determination.
In the case of fission fragments it
is
known (14) that
the most probable fission fragment pair which results
from this fission process has the following characteristics:
Light fragment
A = 97
E(s
1
Heavy fragment
A = 138
) = 95Mev
E(si ) = 67 Mev
z = 54
z = 38
If the energy deposition rates of these two fragments
are compared over the track length from s=O to sa=ar*
( this is
the portion where the largest dE/ds will be
obtained) then it
is
seen that the lighter fragment
has the higher dE/ds, and will give a higher aE.
42
Therefore, for the threshold condition, only the light
fragment need be considered.
Thus, for fission fragments
the initial energy is known.
The initial energy of the PKOA must be calculated.
Since the energy of the neutron is
known, the energy
of the PKOA can be calculated from elastic scattering
theory (15):
E(O)PKOA
where
=
En 1
[I
A- 2
(- A+/
(1-cos6B)
En is the neutron energy
A = atomic weight of the PKOA
t = the angle the neutron is scattered in
the center of mass frame of reference.
For a maximum E(O),
This is
cos 0 is
taken as zero.
the only important difference in the
computational procedure for the threshold values.
(2.37)
43
Chapter 3
Experimental Apparatus and Procedures
3.1
Apparatus Modifications
The experimental apparatus was the same as that
used by Bell (5) and Tso (6), with a few modifications.
The original test chamber is shown in Fig. 3.1.
The test bubble, X , is suspended in the suspension fluid,H,
(Dow Corning 550 fluid for the case of water and propylene
glycol) and covered with the cover fluid I ( Nujol, a
heavy mineral oil, for the case of water and propylene
glycol ). The test chamber, A , is a cylindrical ( 10"x 3"
diameter) steel chamber. The test bubble is visually
observed through the window B . The bubble temperature
is recorded by two Chromel-Alumel thermocouples ( K and L).
The chamber is heated by two electrical heaters (Mand N).
The cover fluid is introduced into the test chamber
from reservoir E through the cover fluid outlet tube J .
The suspension fluid is introduced from the bottom of
the test chamber from the reservoir Q. There are two
cooling devices in the test chamber. One is the condenser,
F , which condenses any water vapor in the air space
left by previous boiling. The condensate is collected
in a container and discharged through a tube into
reservoir R . The other cooling device is the convection
generator G . This serves not only as a cooling device
44
F/1Q. •.1
45
Number 1 to 10
valves
A
boiling chamber
B
observation window
C
light window
D
bubble entrance
E
cover oil reservoir
F
condenser
G
convection generator
H
supporting oil
I
cover oil
J
cover oil outlet
K
top thermocouple
L
bottom thermocouple
M'
bottom heater
N
top heater
O0
air pressurizer
P
pressure gauge
Q
supporting oil reservoir
R
waste tank
S
flow indicator
T
supply water line
U
convection generator cooling lines
V
condenser cooling lines
W
neutron source
X
water bubble
Y
fibre-glass insulation
46
when the chamber is being cooled down, but it serves the
important function of keeping the test bubble centered
during the experimental run. Since this convection
generator is such an important part of the apparatus,
it is shown separately in FIG. 3.2 . The manner in which
the bubble is kept centered is also shown in this figure.
The cooling water through the convection generator coils
causes a convection current which circulates the supporting
fluid such that the flow is radially towards the top of
the generator, down through the generator, and radially
out at the bottom. In this way the bubble is kept radially
in place in the test chamber. The vertical position of
the bubble is set by adjusting the supporting oil level.
Two important modifications were made. The first,
and most significant, was a change in the method of
illuminating the test chamber. In the original design a
high intensity lamp was placed above window C, and this
was considered adequate. It provided sufficient light to
see the test bubble, but it was of considerable strain
to the eyes, thus affecting the ease and accuracy of
obtaining the data, since all data was visually obtained.
The modification which was made placed the light source
directly into the chamber, giving a far better illumination
of the test bubble. FIG. 3.3a shows the original method,
and FIG.3.3b shows the final illumination method. ( Only
the section of the chamber which supports the lighting
system is shown in FIG.3.3 )
47
CONVEC 7O/A
\ITHER•loc oUPL E
WA TER
PR oBE
COA/VLC TI/OA GEzNERA TOFR
F/G. 3. Z
48
LAMP
TEBS 7"
s 11a/.-L
F/G. 3.3ca
TES
T
~TiB E~C
la//
.sougc46
AFI G. 3. 3b
49
This modification required a great amount of work.
Since it was not desired to construct a new chamber, it
was necessary to work through the existing window C. This
created a number of problems, the greatest being that of
obtaining a pressure seal at window C. The problem was
eventually solved by using the combination of metal
rings and hard rubber gaskets to sandwich in the test
tube flange. (see FIG.3.3b) Pressures of up to 100 psig
were obtainable with this design.
The other modification was to introduce a mirror
system for the observation of the bubble. The original
design required the experimenter to place his eye directly
in front of observation window B. This was an extremely
uncomfortable position, and had to be maintained for the
length of the experimental run which often exceeded
a half-hour.
The mirror system (shown in FIG,3.4) permitted the
experimenter to observe the bubble from a distance, and
made it easier to monitor all gauges and recorders
indicating the temperature and pressure of the test chamber.
The other equipment which should be mentioned is
the radiation source, a 5 curie Pu-Be source, and its
associated shielding.
The source itself is a small cylindrical element, which
is contained in a rectangular aluminum container. This
is attached to a masonite block. This entire unit is
kept in a shielded storage container until the source
TEST
Cl-/AM//
BEi~
/v/Rj'iOR
uv5alqvn lo/v
F/G . 3.
WINDOW B
51
is needed. It is then placed in a shielded housing
which has been constructed adjacent to the test chamber.
( The source and housing are shown in FIG. 3.5 )
The pressure gauge P was tested for accuracy with
a dead weight tester, and it was found that the gauge read
approximately 3 psi too low. The calibration curve is shown
in FIG.3.6 ,
3.2
Experimental Procedure
The procedure for a typical experimental run will
be outlined to describe the operational procedure.
Before beginning any experiments with a new test
liquid, it is important to thoroughly clean the inside
of the test chamber. (Washing with water and then with
acetone is recommended.).
The operational procedure is outlined below:
1) Introduce the suspension fluid by opening
valves 6 and 10, and closing all others. This
will force the suspension fluid from the
reservoir Q , into the test chamber. The
supporting fluid level should be brought to
near the top of window B (as shown in FIG.3.1).
Once the desired level is reached, valves 6
and 10 are closed.
2) The cover fluid is next let into the chamber
by opening valves 1,2, and 10. Approximately
52
MA SoV/ 7e 8L OC /
I
M]
Al
COn ToA IAEkR
FIG. 3. S. 4
P
HOuV/lG
SH/•L DED
TECTHA
cH 4Me/?
A/-G. 3. .4b
CA L/BRA TI-ONN
CURVE
FO/?
PREissURE
GAUGL-
lo
go
0,
vl
60
/0
20
30
+0o
DiEA
'
6O
WEY&IT
F/c4 3. o
7
TESTER
JO
/00
(p51)
an inch of cover fluid is sufficient. When the
desired level is reached, valves 1,2, and 10 are
closed.
3) Valve 8 is opened to allow the convection
generator to start flowing.
4) The test bubble is then introduced through
opening D , by means of an eye dropper. The
test bubble is placed above the center of the
convection generator.
5) After the bubble has been introduced, opening
D is closed, and all valves, except 8, are
closed.
6) Valves 9 and 10 are then opened, with valve 10
being adjusted to the desired pressure.
7) Heaters M and N are turned on, and the temperature
is constantly monitored through thermocouples
K and L
8) Once nucleation is observed, or the run is to
be terminated, both heaters, M and N , are turned
off, valve 7 is opened, and valve 8 is turned
on full (if it were not already fully opened).
The chamber is then left to cool down. The
time required for it to cool down depends on
the final temperature which was reached during
the heating process. A final temperature of
500 F will require approximately 45 minutes.
55
9) When the chamber has cooled down the top layer
of liquid must be drawn off, since the test
bubble will have broken up and must, in any case,
be removed and a new test bubble put into the
chamber. This is done by raising the liquid
level (by opening valves 6 and 10) until all
bubbles are above the opening of the cover oil
outlet tube J. All valves are then closed.
Valves 9 and 10 are opened, thus pressurizing
the chamber. Then valve 5 is opened. This will
draw off the top layers of fluids and eject them
into reservoir R .
When this has been completed all valves are
closed, and the system is ready for another
experimental run, beginning with step 1 of this
procedure.
3.3
Experimental Difficulties
While running the experiment a number of problems
may arise which can create serious difficulties, leading
to a premature termination of the experiment.
One problem to be avoided is that of mixing of the
cover and suspension fluid. This can occur if valve 6 is
opened too quickly or too much, causing the suspension
fluid to shoot up into the cover fluid. Also, if reservoir
Q is empty and valve 6 is opened, air will shoot throug'h
the 1lquiid atid cause violent
•g•t. Liorn ad
mixi•
4f' the
fluids. The physical properties of the cover and suspension
fluids are such that once they are mixed they form a
cloudy solution and are very difficult to separate.
When mixing occurs the chamber must be emptied and
cleaned, and new fluids introduced. It is therefore
important that reservoir Q always be filled. The same is
true for reservoir E .
Another point to remember is that when cooling down
the chamber after a high pressure run, the pressure must
not be suddenly reduced. This will cause immediate and
violent boiling which will thoroughly mix the fluids.
The chamber should thus be cooled down to a temperature
below the boiling point of the liquid at atmospheric
pressure, and then the pressure can be released.
Another problem might be a flow blockage problem
with the convection generator. The tubing used in forming
the convection generator is a very small diameter
refrigeration tubing. Ordinary tap water is used as
the coolant water, and this contains a small amount of
chlorine which tends to corrode the inside of the tubing.
There are a number of bends in the small diameter tubing
which can become blocked due to small particles already
existing in the water or produced by corrosion. If the
tubing is not periodically cleaned (a weak solution of
sodium hydroxide is recommended) the corrosion deposits
will soon build up and block the flow. This was the case
in the present work. The corrosion deposits were too
.27
thick to be able to be cleaned, so a new convection
generator had to be constructed.
During the propylene glycol experimental runs, a
change in color of the test liquid was seen to occur
at around 300 F. This change is believed to be a chemical
characteristic of the liquid. (This characteristic is
also seen in other organic liquids.). Also, small dark
particles were noticed in the test chamber after a
number of experimental runs had been made with propylene
glycol. These are believed to be due to residual uranium
nitrate particles which remained in the chamber after the
propylene glycol test liquid boiled. (The uranium nitrate
was introduced to the test liquid for the purpose of
providing fission fragments. See section 3.5 for a further
discussion on this matter.) These particles required
additional cleaning of the chamber.
3.4
Indication of Nucleation
In all previous work on radiation-induced nucleation,
the actual production of vapor bubbles in the test liquid
proved the nucleation event. In the experimental procedure
developed by Bell(5), nucleation is observed through
the jumping or break-up of the test bubble. That is,
when the bubble is seen to suddenly jump, nucleation has
occurred.
For fission fragments at low pressures there is
no difficulty at all in observing when nucleation occurs.
58
The nucleation event at these conditions causes the
bubble to actually break up or "explode". At high pressures,
nucleation due to fission fragments will cause the
bubble to jump suddenly. This is easily seen since it
is a fairly large jump. The first jump is the important
one, since this gives the minimum superheat value. If
the temperature is raised beyond this point, the jumps
will be more frequent and become more violent.
For the case of fast neutrons it is more difficult
to observe the nucleation event. To begin with, higher
temperatures are required, which immediately lengthen
the experimental run. At low pressures the nucleation
event causes the test bubble to jump in a manner similar
to the case of fission fragments in the test fluid. But
at high pressures the test bubble jumps only very slightly,
and extremely careful observation is required in order
to notice this small movement of the bubble. This fact
makes it difficult to obtain data for fast neutroninduced nucleation at high pressures.
3.5
Fission Fragments in a Test Liquid
Fission fragments are introduced into the test
fluid dy dissolving a minute quantity of Uranium Nitrate
( U0 2 (N03 )2 -H2 0 ), in the test liquid. The amount of
Uranium nitrate dissolved can vary. For the case of
water, the concentration used was t 0.0087 gm of uranium
nitrate to 1.0 gm of water. For the case of propylene
59
glycol , 0.0029 gm of uranium nitrate to 1.0 gm of test
fluid was used.
The fast neutrons from the 5 curie Pu-3e source
will cause fission events in the uranium nitrate, thus
creating the desired fission fragments.
60
Chapter 4
Fluid Selection
4.1
3ackground Information
As mentioned previously, the experimental technique
employed by Bell consisted of suspending the test liquid
in two other fluids. This gives the important advantage
of having the test liquid free from any surface effects.
The test liquid used by Bell(5) and Tso(6) was water.
The supporting fluid was a Dow-Corning 550 silicone
fluid. The cover fluid was a heavy weight mineral oil,
which has the trade name of Nujol. Both experimental
and analytical work was done with water.
Tso(6) did some preliminary calculations for benzene
as a test fluid, exposed to Pu-Be neutrons. A literature
search found benzene(4), acetone(4), and diethyl ether (2)
used as test liquids in radiation-induced nucleation
studies. It was thus decided to confirm and extend
the work of these other investigators, and to obtain
analytical results using Bell's(5) constant "a" theory.
4.2
Test Liquid Properties
A number of physical properties are required for the
computation of the superheat temperature. These properties
are listed below:
61
1.,)
Vapor Pressure
2.) Liquid Density
3.) Vapor Density
4.)
Surface Tension
5.) Heat of Vaporization
Data for each of the above physical properties must be
known at a number of temperatures, since the calculational
procedure requires an empirical relationship between
the physical property and the temperature.
This empirical relationship is obtained by feeding
the data for various temperatures into a polynomial
regression computer program. This program will give
an expression of the forms
P = A + BxT + CxT 2 + DxT 3 + .......
where:
(4.1)
P = the value of the physical property
T = temperature at which the property is to be
evaluated
A,B,C,D,....
are constants determined by
the program.
The polynomial regression program is given in
Appendix C.
In addition to benzene, acetone, and diethyl ether,
a search was also conducted to find the required properties
for methyl alcohol, ethyl alcohol, nitromethane, and
propylene glycol.
62
All required properties were eventually found, and
the empirical relationships,along with graphs of the
physival property versus the temperature, are given in
Appendix A .
4.3
Compatibility Requirements
A simple schematic drawing of the configuration
of the liquids is shown in FIG.4.1 below;
TEST BUBBLE FLOAT/I/G
/M,
-L"SPENSIOAJ
FLUID
FIG. 1.
/
The three fluids must exhibit the following characteristics:
1.) The three fluids must be immiscible with
each other. That is, they cannot be soluble
with each other.
2.) The relative densities must be such that
the supporting fluid has the highest
density, the cover fluid the lowest,and
63
the test fluid density must lie between
the two.
3.) The boiling points of the cover and
supporting fluids must be greater than
the boiling point of the test fluid. The
lower the test fluid boiling point, the
easier the experimental work.
4.) The surface tension difference between
the test fluid and the other two fluids
must be fairly large, with the test fluid
having the largest surface tension value.
The fourth factor listed above is not that obvious
at first, but it is one of the most important factors
affecting the choice of the three liquids.
4.4
Difficulties in Fluid Selection
Benzene and Acetone were originally planned to be
used as the test fluids, since some work had already
been done on these fluids (4). But benzene and acetone
were not able to be used with this three liquid
configuration. The problems presented were twofold:
1.) The densities of benzene and acetone are
very low:
- benzene density
-acetone density
0.87901 gm/cm 3
0.78998 gm/cm 3
(20°C)
(20"C)
64
2.) Benzene and acetone are two of the most
universal solvents in use. In other words,
they are soluble in almost any liquid.
These two factors seemed to work together. The few
liquids which are insoluble with benzene or acetone
all
have higher densities. This eliminates any possibility
for a cover fluid, which must have a lower density.
Evidence of this solubility problem can be seen in
any solubility chart. TABLE 4.1 is a small section of
a large solubility chart ( reference (17) pgs. 58-60 ),
and it is seen that there are indeed few liquids which
are insoluble in benzene and in acetone. The liquids which
were found to be immiscible with acetone and benzene are
listed in Table 4.2 and 4.3 , along with their density.
In all cases, those liquids which were immiscible with
benzene or acetone had higher densities . An extensive
literature search ( and consultations with graduate
students in the organic chemistry department at M.I.T.)
failed to produce a suitable cover fluid for either
benzene or acetone.
Therefore, only analytical work was able to be
done with benzene and acetone. However the data of
El-Nagdy (4) is later compared with these analytical
results to determine the "a" value predicted by this
data.
65
ACETONE
BENZENE
M
Acetone
M
Isoamyl acetate
M
Benzene
M
Chloroform
M
M
Diethylenetriamine
M
M
Ethyl alcohol
M
M
Ethyl Benzoate
M
M
1,3-3utylene glycol
M
I
Ethyl ether
M
M
Methyl isopropyl ketone
M
M
Triethylenetetramine
M
M
Glycerol
I
I
Propylene glycol
M
I
Trimethylene chlorohydrin
M
M
Carbon tetrachloride
M
M
Nitromethane
M
I
Diethyl ether
M
M
Pyridine
M
M
Diethyl cellosolve
M
M
Triethyl phosphate
M
M
M = miscible
I = immiscible
TABLE
4.1
66
TABLE 4.2
Liquids immiscible with benzene
Liquid
density gm/cm3 (20'C)
1,3-3utylene glycol
1.005
3-chloro-1,2-propanediol
1.326
ethylene glycol
1.1135
glycerol
1.261
1,2-propanediol
1.036
1,3-propanediol
1.053
diethanolamine
1.089
formamide
1.133
hydroxyethyl-ethylenediamine
0.980
nitromethane
1.1138
TABLE 4.3
Liquids immiscible with acetone
Liquid
glycerol
density gm/cm3 (2000)
1.261
67
A search was conducted to find a test liquid other
than water, and supporting liquids which would be
compatible with it.
This is a difficult and time
consuming procedure, since there are four independent
variables which must be taken into account (miscibility,
density, boiling point, and surface tension).
The easiest method of search is to begin with the
miscibility condition. This is the quickest way to
eliminate most of the unsuitable liquids. The first
step would be to choose liquids which are immiscible
with a large number of other liquids. ( The miscibility
charts given in references (17)-(21) are recommended for
this purpose.)
After obtaining a number of candidates from the
miscibility criteria, the boiling points should be
checked. The test liquid should have as low a boiling
point as possible. ( Reference (22) gives the boiling
points of most of the commonly used liquid solvents
conveniently in order of increasing boiling point.)
The possible choices are then checked as to densities.
The test liquid must have an intermediate density.
( Reference (22) also gives the densities of a large
number of liquids in ascending order.)
In addition to the organic liquids there exist
some light weight mineral oils which might be able to
be used as cover and stopping fluids. These are for the
most part clear liquids with high boiling points, and
68
are immiscible in many organic liquids. But even though
they are referred to as light-weight oils, their densities
are not much less than that of water. ( Nujol has a
density of 0.886 gm/cm 3 at 20 0C, which makes it heavier
than either benzene or acetone.)
Upon completion of the preceeding analysis, a
number of possible choices were available. The exact
effect of surface tension was not known at this time.
Small quantities of the most likely candidates were
obtained, and compatibility tests were made. These
compatibility tests simply consisted of placing the
supporting and cover fluids in a beaker ( a similar
arrangement as in the test chamber) and then introducing
the test fluid. This is an exact simulation of the
manner in which it is done in the test chamber.
Table 4.4 gives a partial list of the results
observed from the compatibility tests described above.
In many cases there resulted what was termed a "spreading
effect", where the test liquid did not remain in the
shape of a bubble, but spread out over the supporting
fluid. This effect is shown in FIG. 4.2.
I
COVER
FLUID--
&vp PPRTVWG
FL V4l
FIG . 1.2
-
TE5 T
fL.I
P
69
SUPPORTING FLUID
TEST FLUID
COVER FLUID
( surface tension in dynes/cm at 20*C )
(1)
(2)
Glycerol
Nitromethane
Nujol
63.3
37.48
23.0
Ethyleneglycol
nitroethane
32.66
Nujol
46.49
Glycerol
(3)
Glycerol
1,2-dichloro- nujol
ethane
propyleneglycol
(6)
Test bubble
slightly
soluble when
heated.
Bubble does
not remain in
bubble shape,
spreads.
Glycerol
salicylaldehyde
ethyleneglycol
test bubble
soluble when
heated
DC FS 1265
nitromethane
Nujol
test bubble
slightly
soluble when
heated.
propyleneglycol
Nujol
Stable bubble
fluid
26.1
(7)
Spreading of
the test fluid.
Nujol
36.9
(5)
Spreading of
the test fluid.
23.0
32.23
(4)
RESULT
DC 550 fluid
23.50
TABLE 4.4
70
The surface tension effect is apparent from these
results. In the fourth test listed the supporting
fluid, glycerol, has a high surface tension (
6 3.3dynes/cm
at 200 C), while the test fluid, propylene glycol, has
a much lower surface tension (36.9 dynes/cm at 20 0 C).
The result of this test was the spreading of the test
fluid over the supporting fluid as shown in FIG.4.2
In the seventh test listed, the high surface tension
glycerol was replaced with a low surface tension
( 23.5 dynes/cm at 200 C ) DC 550 silicone oil. The
result of this test was the successful configuration
shown in FIG. 4.1 . In both of the cases mentioned the
surface tension difference was large, but for a
successful configuration it is not only necessary that
there be a large difference between surface tensions,
but also that the test liquid have the highest surface
tension.
In the experimental work with water as the test
fluid, the water test bubble showed exceptional stability
in the supporting (DC550 fluid) and cover (Nujol) fluids.
The surface tension of water is found to be 72.0dynes/cm
at 20 C . This tends to support the conclusion arrived
at above.
Thus a compatible
cover fluid :
test fluid :
set of fluids was found:
Nujol
propylene glycol
supporting fluid : Dow Corning 550 fluid.
71
Experimental work was then done using this test fluid.
( Due to time limitations, the search for more compatible
fluid sets was not continued to find another possible
test fluid.)
4.5
Discussion of Surface Tension
Since the surface tension plays such an important
role in the selection of a compatible fluid set, a few
words on the nature of surface tension are in order.
Surface tension (U-)
is usually defined from a
molecular viewpoint. An individual molecule in a body
of liquid is subject to forces due to its neighboring
molecules. ( Theory indicates that these are electrical
forces (23) ) These forces will fluctuate due to molecular
agitation, but when averaged over a finite period,
their effect will be zero, so that a molecule can
move about in a liquid without doing any work against
these forces (24). In other words, the resultant
average forces are equal in all directions inside a
body of liquid.But when a molecule comes to the surface
of this body of liquid, the molecule must ultimately
move against unbalanced forces, since it will no longer
be surrounded symmetrically by other molecules. At the
surface, therefore, there will be an uneven distribution
of forces acting essentially normal to the free surface
of the liquid. ( This will be normally inward, since
the electrical force is essentially an attractive
force (25) ). There is thus a pull toward the interior
72
of the body exerted on the surface layer.
Since this surface force is uniform due to the
averaging effect of the large number of molecules
per unit of surface area, it tends to produce a surface
with the smallest possible area. ( If the liquid is
free from gravitational effects, a spherical shape
would result.) This inward normal pull is usually
defined as the surface free energy. And to extend
this surface, work must be done equivalent to this
energy. This is usually referred to as the surface
tension.( This term is not totally correct, but the
idea of a tension in this free surface of a liquid
is familiar as an explanation of the tendency of a
liquid surface to assume the form having a minimum
area.)
The units of surface tension are: dynes/cm ,
lbf/ft , or ergs/cm 2
Relationships between the surface tension and
other physical properties will now be considered.
The first important fact is that as the temperature
of the liquid is increased, the surface tension decreases.
This is due to the fact that the increase in temperature
gives the molecules in the liquid more energy, and thus
larger fluctuations, and, since the attractive forces
are close-range forces, they will decrease with an
increase in the distance between molecules. At the
critical temperature the surface tension is zero.
T7."
A number of relationships are given between the
temperature of a liquid and its surface tension. These
are all of the general form
= A - BxT
(4.2)
where:
-
T
-
surface tension
=
temperature at which the surface tension
is to be calculated
A and B
are constants which vary with the liquid.
There is also a relationship between the viscosity and
the surface tension. An equation relating these two
parameters has been formulated by Buehler (27), and is
given below
=2a
(4.3)
= surface tension
wheres
= viscosity in millipoise
7/p
= a constant which varies for different
liquids.
74
A relation between surface tension, density, and
temperature has been given by Macleod(34) :
-
-
= constant
(4.4)
(D-d)4
where;
TP = surface tension
D
= liquid density
d
= vapor density
Very recently there has been developed a correlation
between surface tension and the dielectric constant,
and surface tension and the index of refraction (28).
The relation between the surface tension and the dielectric
constant is shown in equation (4.5)
% /6 [E
/)
where:
0
Cs
(2Es
u)1
(4,5)
= surface tension in ergs/cm 2
= static dielectric constant
Equation (4.5) is good for all liquids with zero dipole
moment. These include: hexane, octane, benzene, p-xylene,
carbon tetrachloride, p-dichlorobenzene, CS 2 , H ,
2
N2
, 02 , A , Br 2 , C12
*
75
For molecules with non-zero dipole moments, the
relation is of similar form, but includes the index of
refraction, 7
, instead of the dielectric constant.
This correlation is shown below:
o VG
hý
nat
r
za-
(49.6)
The previous correlations between surface tension
and other physical properties ( viscosity, density,
dielectric constant, index of refraction) are very
useful, since surface tension data is difficult to
find at more than one temperature.
The previous few paragraphs have been an introduction
to the concept of surface tensions. For a more detailed
discussion on this topic, references (23) to (28) are
recommended.
76
Chapter 5
Results
5.1
Analytical Results
Analytical results have been obtained using the
energy balance method of Bell (5). The results include
fast neutrons in the test fluid and fission fragments
in the test fluid.
For the case of fast neutrons in the test fluid,
the initial energy of the PKOA was 2.12 Mev. This
initial energy value is listed on the graphs as
"EPKA"
(energy of the primary knock-on atom).
(The
value of 2.12 Mev for this initial energy was obtained
through an analysis by Bell(
(5) pg. 173) considering
the Pu-Be neutron source and its associated energy spectrum.)
The test fluids for which analytical results
were obtained are:
1.) Water
(FIG.5.1 and FIG.5.2 )
2.) Propylene Glycol (FIG.5.3 and FIG.5.4)
Benzene
( FIG.5.5 and
FIG.5.6 )
4.) Acetone
( FIG.5.7 and
FIG.5.8.)
3.)
5.) Nitromethane
( FIG. 5.9 and
FIG.5.10 )
The results are given in the form of a graph of
the threshold superheat ( amount of superheat ) versus
the system pressure, for a number of "a" values.
FAsT NEUTRONS
EPHA
IN
LA7ATER
2,./
/= A/e V
/86
/60
/60
-=G. o
(1=7.o
,
/00
8. o
. = 12. 0
80
GO
20
'0
60
80
/00
/20
/IO
160
PREssURE (Ps/A)
/80
200
220
2i-O
260
Z80
Z80
300
300
320
320
FC/SS/ A/ P-A-G
•/Aw TS
IN
wi r-crI
A-r.. )
70
G0
so
30
3o
2o0
/
20
J
3o
4
P (PS) ]
.s
6l
70
80
9
79
r ml/
F4ST
TRROAS
I/V
PROPYL-EN-
GLYco0L
Me v
EPKA = 2 /2
=-/00o
-
C, = ,.0
4-= 3.o
o
c =-".
/0
20
30
o0
4/O
PA'gSS £fl?
60
(ps/o)
FI/C. .3
70
o0
pO
/0o
80
F/SS/ION FRA GMEV TS
A~~_-·,·
/0
20
30
90
d
· C··-
$so
PRESSuRE
0
(P/1,4 )
F/ C. < A/
t
h
70
•
8o
go
/00
81
FAST /NEU/TR OVS
IV
BEAI/Z E NE
Me V
EPH/A = 2. /
//o
/o0
K
a =2..0
3.0
-*
0
Ll
'
_I
2o
30Y0
93
O
CO
PrFrSSJUIE (PS/A)
FIC. I
70
Bo
9)
/AIp
82
FISSI/ON/RAG
ME T 5
aX
2.0
y.o
/0
20
30
90
60
do
PREssun~-
F/c. S56
(P~A)
70
ed
90
/00
FA S T NE UTROANS
ACETONE
ePra = 2./ 2 /e v
I-.
/,o
"A
2.0
a}
3,o
.o
/O
zo
30
4o
Slo
6o
PRECSSU cR (Ps/A)
FIG J.S 7.
70
eo
9o
/~~0
/Oep
84
E;/SS/OA/ FR9A G6I/MEN TS
ACE TOa/VE
3,0
4(0
/0o
2D
30
,
'I ro
PRESSURE
F/1 . •8
(0
(Ps/ A)
70
o
90
/oo
85
1FA S T
A/E / TRoNS
/E T
MIT"R OME T-HA IVE
E PfA = 2 12 Me v'
/3X
a
= /. o
6.0
7,0
40
/o
2o
30
'Yo
PR, SSU
s
40
r0
(/E A
(p/,A)
-/C. c. 9
70
po
o9
/00
86
F/SS/AV
A /TTf-rn
t-R7AGME7NTS
//I
A.w-
-r i/
A
Af r-
17%
N
So
30
9
-O
Pit9SSURE
ric.
5
60O
(Ps/A)
/0
70
So
?a
/0o
87
5.2
Experimental Results
Experimental results were obtained for fission
fragments in water and propylene glycol.
For water, fission fragment data was obtained
at four pressures: 14.7psia, 32.7psia,
53.7psia, and
74.2psia . These were the same pressures as were
tested by Tso(6), since one of the objectives of this
work was to attempt to confirm the results obtained
by Tso for the fission fragment case.
The results obtained in this work are shown
on FIG.5.11.
These results and the results obtained
by Tso are listed together in TABLE 5.1, shown below.
TABLE 5.1
PRESSURE
AMOUNT SUPERHEAT
(psia)
TSO (6)
( OF
OBERLE
14.7
63.5 f 2
32.7
49-3
53.7
38.7 a 2.5
39.2 ± 4.1
74.2
34.2 + 2.4
33.3 ± 2.*
As is
seen above,
,
3
62.4
j
2.5
48.5 j
2.0
the results obtained in this
work agree quite well with those obtained by Tso (6).
FSS/O
T s
P/
iRAGME
EKPEA'/A1A 7~4L
/,V
WA 7?R
Esuv. TS
K
LI3
=-3.0
'.
7.0
/0
2o
30
9.o0
I
60o
PREEsSURE
FIG.&. //
70
(PsI4)
4oO
90
/l
89
For the case of fast neutrons , only one data
point was obtained. This was for the case of fast
neutrons in water at 55 psia. A large number of
experimental runs were required to obtain one good
result for this case. This was due mainly to the fact
that this data was at high temperatures , and the
convection generator was not very effective in keeping
the bubble centered. Therefore, due to the difficult
and time consuming experimentation involved, further
work was not done with fast neutrons, since work was
still to be done with organic liquids.
The result obtained for the single fast neutron
result is shown in TABLE 5.2, along with the results
of Tao (6) at this same pressure.
TABLE 5.2
PRESSURE
AMOUNT SUPERHEAT
psia
("F )
TEMPERATURE
RAMP (OF/mi n)
OBERLE
55
105.0
2.0
TSO (6)
55
55
102.4
1.6
1.e35
it
105-.9
Bell (5) and Tao (6) also encountered considerable
difficulties with the fast neutrons in water. Tso ,
( (6) pg. 68 ) has mentioned the difficulties and
reliability of their work.
90
The experimental data obtained for propylene
glycol exposed to fission fragments is shown in FIG.5.12.
The superheat data was obtained at four pressures:
14.7psia, 20psia, 40psla, 60psia.
Considerable difficulty was encountered for the
higher pressure ( 40psia and 60 psia) since high
temperatures (around 500 0 F ) were required, and the
convection generator was not able to keep the bubble
centered. Due to the convection generator problem, more
than one bubble was placed into the test chamber for
each experimental run, so that if one or more bubbles
drifted out of sight, there would, it was hoped, still
remain at least one test bubble in sight. This was not
always the case, since very often all of the test bubbles
drifted out of sight. There were not as many good experimental
runs for these pressures as there were for the two lower
pressures studied. Due to this fact, and due to the presence
and movement of a large number of test bubbles per
experimental run, the data obtained at these higher
pressures is not considered to be as accurate as that
for the lower pressures.
5.3
SensitiLvty Study of superheat IData
The minimum superheat conditions are especially
sensitive to the surface tension of the fluid and to the
energy deposition of the charged particle. It was
91
F/.SSOA/ FRAGMENTs
/IA
GLYCOL
PRI Pyt/LE
EX P~R~MGtV7A4 L
R5S ULTS
OBERL E:
$
? 44
bV;
J
/o
20
3
yo
9
.o
P/?OIssu
/ /G.
60
(Ps/A)
1./2
70
o
~O
/o
92
reported by Connolly (33) that variations in these
two factors significantly changed the analytical results
of his group's work.
Therefore, a sensitivity study was made for water
and propylene glycol exposed to fission fragments.
For the case of water, a ± 7% variation in
the surface tension was studied. The result on the
minimum superheat is shown in FIG.5.13
A 20% variation in the mean charge of the radiation
particle was considered. ( The mean charge is the most
sensitive factor in the energy deposition calculation. And,
as suggested by Bell (5),
a 20% variation is conceivable.)
The effect of this large uncertainty is shown in FIG.5.14
It was seen that a positive uncertainty in the
surface tension increased the minimum superheat, while
a negative uncertainty decreased the minimum superheat.
In the case of the mean charge a positive uncertainty
decreased the mean superheat while a negative uncertainty
increased it.
Therefore, when these two uncertainties
were combined to obtain a maximum uncertainty in the
minimum superheat, they were combined in the following
manner:
maximum positive uncertainty---
+7% in surface tension
-20% in mean charge
maximum negative uncertainty--
-7% in surface tension
+20% in mean charge
The result of the maximum uncertainties are shown
in FIG.5.15 .
93
,/ss /o/V
t7Z
-
#
F,4G•A'V TS
//v
BUCC'RrA/ 74Y
/1A
SERL E
WA T76F
Sl1RF4CeCI
TEAIS ,aOv
A4A-rA
-7o/
+7%
- a =3.0
S= . o 7
4O
to
3
4/o
so
ro
PeSSRE' (PS/A)
FIG. . 1/3
70
o0
5o
/lo
94
EI/SSIOV
-RW
CMEA/
rs
/w/ WA TE/7
/1 C FR TA/1w 7T/
OBEPL.DA
E
TA
-2o0
/o0
o
30
+P
So
(o
PIfESS/R E
FIGS..
i4
(Ps /A)
70
g0
9P
/0o
95
IERAGAEN 7 S
F/SSI/OON
/,A
WA 7T R
ULIC Efi? TA /A/ T/E S
CO/1B8/4'ED
fAIA/v
CcYA/?
FAA/A
SOBERL -2
c#ARGE
OB R L K DA TA
· 77: -~o,<)
So
I
9o0
75~
/o
2o
30o
fdp~a
0o
,o
PR-irswSRE
FIG
/
0o
(PsCPA)
/6
70
So
9o
/oo
96
This same type of sensitivity study was made for
the case of propylene glycol. The percent variations
used were:
maximum uncertainties in surface tension--
L5%
maximum uncertainties in mean charge --
: 20%
The directional affect of each uncertainty is the same as
for water. The results of these studies are shown in
FIG.5.16, FIG.5.17, FIG.5.18 .
5.4
Comparison with other results
Experimental results have been obtained by
other researchers.
Deitrich (3) has obtained data for the case of
fission products in water. His results are shown in
FIG.5.19, along with the results from this work.
El-Nagdy (4) has obtained experimental data for
Benzene and Acetone exposed to fast neutrons at atmospheric
pressure. The fast neutrons were at different energies than
those from the Pu-Be source used in this work. His
experimental data is
shown with the analytical results
from the energy balance method of Bell(5). (Adjustments
had been made to conform with the neutron energies used
by E1-Nagdy.)
The results are shown in FIG.5.20 and FIG.5.21.
( The 14.1 Mev neutrons correspond to an EPKA of 4.0 , and
the 2.45
Mev neutrons correspond to an EPKA of 0.695 Mev.
97
F/SS/o/vN FRAGG9sTTS
po
GL ycoL
PPy ,v-,
SUlRF,Ac
TfdA/lS/olv
EXPErIMEIVTAL
DATA
(OBERLE)
s0
,
T34i
-fs o0
30
Qd:OI
Zo
a.-
/o
2o
30
.foO
60
PRiSSVRE (PS/A)
F1&.X /6
7o
80
90
/oo
Flsss/o/
St 2 0
x-RAf
,G
rJ
Ul•'c dETA/A/T7
/1A"
Al c 4AIc Y4G
Of
f+ZO/t
~3o
I
/9
o
.30
ye
ro
&0
70
(PJOs)
F G.So/7
go
?o
loo
99
//
F/ss/OlV FRA vlGAIM7
P R o Py iyt cLyco0
1/,4e
M
t~~
-~%
/
7A0
MS
F*"
+toz
',o
a,.
.44,
r· ·s~·
41
/.0
/0
6'
f
/o
,o
30
'yo
6o
oo
Go
7o
(Pso)
F/G ./8
e
ý
ta
/o0
g/iRACg7/-S //
N
I
O
O
70
60
6. 0
7.0o
/a.o
bo
·I
I
Ie
4*0
7/.
/o00
/lo
I
0oU
IPRe.ssafa- (P.5/A)
FIG. .:/9
DE/ 7R,/IcH/
WA•
A T7ER
101
4r
/V6•eJtro•o/s
"/u r•TAN J
/' r/ /fee v
ey
9e V
2•.•
/Aw
V6 ,•4o0a,
GP/ :
-A'~A
/,/
Ne V
I&
1%
'I 'BW
/oo
60
60
R0
/O
/0~J
· 10
3
'o
30
yo
P~pEss
o0
~o
~R~-
r Q6. ozo
0
70
(P.S/A-
;
ew
pa
Aga
102
/A/
7'$?AVS
FASf1 AN1J
/,'e
SYs,.•'ev
/4cg
•ATO/V•
r6o,,
EL-•• - A4C
-.
--.
/4 f/ / e/v
*f. 4/$ Lfg&
,2da
\\
\\
/20
~a
'C
"'C
Z~r~c
/o
o
3o
o0
so
O
~o
AFasG/.
</I
FIG. &2
70o
)1
go
/0o
103
Chapter 6
Conclusions and Recommendations
6.1
Conclusions
The experimental results for fission fragments
in water have been shown in FIG.5.11.
These results
can be considered very reliable, since the nucleation
event was very noticeable, and, more significantly,
this data was very close to the data of Tso(6),
who
also did experimental work under these conditions. It
is
therefore reasonable to conclude that these minimum
superheat results for water exposed to fission fragments,
can be considered very accurate and reliable.
These experimental results fall between "a"
values of approximately 3.5 and 4.5. In any case,
the "a" value is less than 6.07,as predicted by Bell (5).
The experimental results of Deitrich(3) have
been plotted over calculated "a" curves. ( FIG.5.19)
These results varied from an "a" value of 10.0, to an
"a" value of 5.0
Deitrich's data also seems to be
fairly reliable, with the uncertainties in his superheat
data of ±t 0.5 F .
Therefore,
the experimental evidence seems to
point to the conclusion that the "constant a " theory
of Bell (5) does not hold for the case of fission
fragments in water.
104
There exists, of course,
is
the possibility that there
some error or uncertainty in the analytical calculations
for the constant "a" curves.
A sensitivity study was made on the parameters
which have the greatest affect on the superheat limits,
and thus on the "a" curves. A +h 7% variation in the
surface tension , and a t 20% variation in the mean
charge was considered for water. ( FIG.5.13 and FIG.5.14)
These two uncertainties were first considered separately,
and then jointly ( FIG.5.15 ). In either case, the
experimental data did not fall within the
a = 6.07 band.
The same type of sensitivity study was made for
propylene glycol exposed to fission fragments.
The
conclusions are similar to those for water. For these
experiments with propylene glycol, all of the data points
were below the
a = 6.07 line. With all of the uncertain-
ties mentioned above, there is a possibility that the
data may all fall within the error band of a single
"a" value. More study must be done on this before
anything conclusive can be said on this. But in any
case, it
can be concluded that the "a" value is not
6.07 as stated by Bell (5).
The analytical results for the fast neutrons are
similar to those of the fission fragments. The difference
being that the fast neutrons require larger superheats.
105
In benzene and acetone ( FIG.5.5 and FIG.5.6 ),
the superheat curves are seen to go downward as "a"
increases, and then to reverse direction and go up as
"a" continues to increase. The reason for this is not
understood at the present time, and further study needs
to be done to explain this behavior.
612
Rayl eigh 's cri terZia
The constant "a" theory of Bell (5) concerned
itself with the break up of a cylinder of vapor. It
was assumed that the vapor cylinder contains disturbances
of all wavelengths. These disturbances are held to be
responsible for the break up of the cylinder.
Rayleigh (35) has developed a theory of jet
instabilities which, in short, shows that there is a
certain wavelength which will
lead to the maximum
rate of growth of the disturbance, and hence break up.
This certain wavelength has been found by Rayleigh
for two extreme cases. In the first case, a vapor jet is
surrounded by a liquid; in the second case, a liquid
jet is surrounded by a vapor . A schematic of these
two cases is shown in FIG6.
.
For the first case, that of a vapor surrounded by
liquid,Rayleigh found this certain wavelength to be
A
=
(12.9 6 )rc
(6.1)
106
/
/
I/j
IjI
I
j
I
I
/
/
/II
/
E
vapor surrounded by liquid
r /
(
/
A
= (12.96)r,
a= 6.07
,,
/
,
,
,/
,
/
t
I
,
--.
/
AL"LAL
,.
',
CASEil
AGT•A• GA
tr
1'
io,
1I-(
I
,
I~~,
ft
' iq u
'
r.
*
(
I'
i
/
.
e
, yJ "
.,.
liquid surrounded by vapor
/ .
./
= (9.02)r,
a =4.79
107
where
A is the wavelength
ro is the radius of the cylinder.
For the case of a liquid jet surrounded by vapor,
the
wavelength is given by:
= (9.02)r,
This wavelength is related to Bell's(5) work
through the relationship
\ = L = ar*
where
r* is the critical radius of the embryo.
a = Bell's parameter
Beginning with the wavelength for the case of a vapor
jet surrounded by a liquid, Bell ((5) pg 200 ) was able
to calculate that the "a" parameter for this case would
be equal to 6.07 .
Similar calculations show that for the case of a
liquid surrounded by a vapor, with
A = (9.02)r, , the
resulting "a" value is 4.79 •
The actual case encountered in the radiationinduced nucleation studies, is believed to be somewhere
between these two cases. That is,
lie between
4.79 and 6.07 .
the "a" value should
( It is probably closer
to 6.07, since the jet is more vapor than liquid.)
108
6.5
Recommendations
Before undertaking any further experimental work,
a number of changes are recommended in the experimental
apparatus.
The major problem encountered was that of the
bubble drifting out of view. The present method of
keeping the bubble centered is by means of the
convection generator which was discussed in chapter 3.
This convection generator, however, was not able to
keep the test bubble centered at all times. It proved
exceptionally difficult to keep the bubble centered
at high temperatures, and many experimental runs
were terminated prematurely due to the test bubble
drifting out of view.
A new convection generator is recommended. This
should have a larger flow-rate capacity and a coolant
inlet temperature as low as possible.
Another recommended modification would be a
separate recorder for each thermocouple. This would
eliminate the switching back and forth on one chart
recorder.
It is seriously recommended that a new chamber
be constructed. This new chamber should include a
larger observation window, preferrably one in which the
entire test chamber is visible. It should also include
a bright and variable intensity light source, either
109
inside or outside of the chamber. But it must supply
direct illumination of the bubble.
A chamber similar to the one used by T.J.
Connolly (31) seems very promising. A sketch of such
a type is given in FIG.6.1a . An alternate lighting
scheme is shown in FIG.6.1b. Both of these cases must
include a convection generator.
The important factors in any new chamber are:
1.) The window to observe the bubble should be as
large as possible.
2.)
The light source should be as near as possible
to the bubble.
3.)A method to keep the bubble in position, such as
the convection generator.
Another improvement which can be made, is the
degassing of the test liquid. El-Nagdy(4) and Connolly
(33)
describe a simple method of accomplishing this.
Since it
is difficult to find supporting and
cover fluids for many test liquids of interest, the
concept of a density gradient suspension fluid should
be investigated as a possible solution to this problem.
Further experimental work should. be done with
more organic fluids, preferrably with those for
which analytical results have already been obtained.
Different radiation sources and types can be
tested, such as gamma rays, or alpha particles.
110
CT
LIGH
c1t0R3a
0
WIavOo
CWAM 3B'i
FIG. 6.2a
'DOw'
50oiRCC
WIMDt
V/E WE
FIG.6.2b
3'"
ill
Theoretical work still
remains to be done.
It does not seem that the "constant a " theory holds,
at least for the cases in which experimental data was
obtained. In any case, it
has been shown that evenif
there is a "constant a " it
is
not equal to 6.07,
but
would be somewhat lower.
A possible area of study which might aid in
understanding this phenomena, is in the area of
the instability and break up of vapor and liquid
cylinders, such as the discussion given by Rayleigh.
112
Appendix A
Physical Properties
A.
I Introduction
Five physical properties of the test liquid
are required for the superheat calculations. These are:
1.)
Vapor Pressure
2.)
Liquid density
3.)
Vapor density
4.)
Surface tension
5.)
Heat of Vaporization
The variation of these properties with temperature
must also be known. A search was made to find tabulated
data of these properties at various temperatures. This
data was then fed into a polynomial regression computer
program, which gave an equation for the physical
property as a function of temperature. The equations
have the following form:
P = a + bT + cT 2
where
dT 3 + ........
(A.1)
P = physical property
T = temperature at which the property is
evaluated.
a,b,c,d,...are constants determined by the
program.
113
The number of terms required depends on the particular
case. The output of the computer program gives the
amount of improvement for each additional term added
to the polynomial. The first three terms are almost
always sufficient.
For some properties a relation between the
property and the temperature may be given directly.
When this was the case, the equation given was used.
The remainder of this appendix will give the
final equations which relate the property to the
temperature, and it will also present graphs of the
physical property versus the temperature over the
temperature range of interest.
Each of the five physical properties have been
found for each of the following liquids:
1.) Acetone
2.)
Diethyl Ether
3.) Propylene glycol
.4.) Nitromethane
5.) Methyl alcohol
6.) Ethyl alcohol
The properties for water and benzene have previously
been reported by Bell (5) and Tso ( pgs. 84-95
(6) ).
The properties are listed in the following manner:
- property involved ( units)
,
( reference source)
- temperature dependent equation
- temperature range over which the equation is valid
114.
A.2
Physical Properties of Acetone
i.) Vapor pressure
log ( Pv) =
( Pv in mm Hg.,
7.23157 -
(22)
T in C)
1277.03
T + 237.23
triple point < T < 190 OC
2.) Liquid density (
(}
Q
lbm/ft
l
3
TOF
)
(4)
= 66.3909 - 0.36738(T) + 1.9336 x 10- 3 (T)2
- 3.25478 x 10- 6 (T) 3
600oC< T230%C
3.) Vapor density ( P lbm/ft 3
e =0.45065 - 8.223510
3 (T)
,
(4)
ToF )
+ 3.8308x10-5(T)
2
6000< T<230°C
4.) Surface tension ( 9 Ibf/ft ,
T*F )
(4)
--= 1.9457 x 10- 3 - 4.8055 x 10- 6 (T)
6000 <T <230 OC
5.) Heat of vaporization (HFG
Btu/lbm , TOF )
(4)
HFG = 251.046 + 0.08274(T) - 0.0017143(T) 2
6000 <T<230 OC
The graphs of physical property vs temperature for acetone
are shown in
FIG. A.1 through FIG.A.5 .
115
v5s TEMPERATURE
VA POR PREssURE
ACETON E,
DIE THYL ETHER
/
/
la
/
gO
J.
/
/
-7
/z2o
/o
/60
/0o0
200
7-
FIG. A.1
22 o
2o
2.60
116
vs TEM PERATUREE
VA POR PENS I/T Y
ii
ACE TON E
DIE TH YL ETHER
i
-
-
17
1.3
.-
/
CL,
o• 0.1
/
0.8
/
0.7
0.6
/
7
O.S4
0.3
0.2
0.i
/~Zo
/Yo
/60
/3o
200
T OF
FI/G. A.2
220
290
z260
280
117
LIQUID DEA/S5IT
/
TE/M PERA TU/R E
vs
ACE TONE
DIE T/HYL E7HER - -
-
N
/Ro
/Y,'
/16
/80
20F
7"OF
FIG. A.3
220
e2'o
e260
118
.SURFACE TEINSION
vs TE/VPERATU R
ACE TONEV
DIE THYL ET ER
17
/C
/3.
II
K::
K
0
IO
N
L
N\
9
8
K
6.
N
K
I
8o
/00
/20
/60
/Io
T
FIG. A.J
i0o
zoo
Z20
Zdo
E
119
HEAT OF VAPORIZATIOAV
vs T'E/vPERATUPRE
AC ETONE
DIETHYL ETHER
2r0
211/7
/oo
2a0
/80
I
/00
/20
f/f
/60
/80
T "F
FIG. A.
200
2Z0
2¾'d
26o
120
A%3
Phyjical ProDerties of Diethyl Ether
1.) Vapor pressure ( Pv psia , TOF )
P,V
(22)
44.2626-0.96031(T) + 0.0056627(T)2
80 0F < T < 300 'F
2.) Liquid density ( •
bm/ft
3
T0 F )
,
(29)
, = 45.6089 - 0.005339(T) - 0.00013836(T)2
65 0 F < T
30 0 °F
3. ) Vapor density ( (,
PV
=
Ibm/ft
3
, ToF )
(29)
2
1.53905 - 0.030649(T) + 0.00013358(T)
65°F< T<300 F
4. ) Surface tension (
-
lbf/ft
TOF )
,
-- = 0.0013942 - 3.81359 x 10
6
(T)
65PF <T < 330 F
5.) Heat of vaporization (HFG Btu/Ibm , TdF )
HFG = 162.277 - 0.00444(T) - 8.31636 x 10
80 F < T<280 0 F
Graphs are shown on FIG. A.1 through A.5
(T)
(22)
2
121
A,4
Physical Pro•orties of ProDylene Glycol
in mm Hg.,
1.) Vapor pressure (P,
log(Pv ) = 8.9171 -
t°C )
(22)
T + 250.7
T + 250.7
OC
O
T <250 0C
2.) Liquid density (
6
in Ibm/ft
3
, TOF )
(29)
= 66.3836 -0.0236(T) - 1.1618x0-5(T) 2
200C <T<280OC
0
3.) Vapor density ( P, in Ibm/ft 3 , T F )
(29)
2 = 0.2627 - 0.002735(T) + 6.650510-6(T)2
20 0C < T< 280 0 C
4.) Surface tension (
V
in Ibf/ft , TOF )
(22)
•- = 0.00285136 - 4.71296x1i-6(T)
<
20oC<T
2 80 C
5.) Heat of vaporization ( HFG in Btu/Ibm , TOF )
HFG = 396.7268 - 0.11884(T)
- 3.7196x10i
(T)2
250C <T< 3004C
The graphs of property vs temperature are shown on
FIG.A.6 to A.10 .
(22)
122
VA POR PRESS URE
vs TMEITPER A TUR E
DROPIPLswe9 GL YC o L
V / TROQME fHA /E
ac
/
Q-i
/
z
29o0
300
320
3410
3~0
T
"F
FIG. A.6
390
101oo
441o
Y60
020o
12j
L/9 uID
DEANS/Tr Y
vs
TEMPERATURE
PROP YL EA/E GLYCOL
/y/ TR/OME T1/A A/E
"
M
f
*N
N-
3Y
T )F
F/G. A. 7
3o60
38 o
oo
124
VAPOR PDENSITY
vs
TE/V/PER-A TL/RE
PRoPYLEt-/E GLYcoL
NITRCo/
"1
//A /V'E
TR
C2,(
0.
O,
-7
0.
2 A-
300
a Z.5
3.o
T 0F
F/ G. A.8
12r
SURFACE TE/SIa'N
/
vs TEA/IPER/A
TURE
PRfPYZ EA/E GL Yc oL
/ T/?o1E7T-/AVE/
-
/8
/6.
1rJ
K
K
K
\
·I
2. Z.
o
27
'300
jo
3AS5
ax<
F/IG. A.9
a7Z5
337
Y,0-
Ieso
125
vs TE/4VPERATURE-
/EA 7T o VA PoRIZA T/1o
PRoP/yIE.
e,
GL YC oL
V / /TRoIAET
7'1AA/E
386
3YO
320/
300
290
ZY4
ZZ4
225
ZSO
27."
300
3 ZS°
3,o
T "F
/ G. A. /O
3 7-
9o o
,
-YSo
127
A. 5
i.)
Physical ProPerties of Nitromethabne
Vapor pressure ( Pv in psia , TF )
(22 )
Pv= 19.1127 - 0.342947(T) + 0.0015267(T) 2
20 C (T < 268 0 C
2.) Liquid density ( Pt in ibm/ft
,
= 74.07513 - 0.0430263(T)
3
,
TOF )
- 2.56308x10i
(29)
5
(T)
2
20OC < T <268°C
3.) Vapor density
( P
in Ibm/ft
3
-0.008079(T) +2.8548x10o
=v
.0.52092
20 0 C T e 268 0 C
4.) Surface tension (
(29)
, TOF )
(T
in lbf/ft , TOF )
(22)
9- = 0.0030398- 6.6158xo0-6(T)
20oC 4 T -268
5.) Heat of vaporization
C
( HFG in Btu/Ibm, T0 F)
HFG = 343.3718 - 0.009790(T) - 2.5489x10
2500c
4
(T) 2
T < 265 OC
The graphs for property vs temperature are found
on FIG.A.6 through FIG.A.10 .
(22)
128
A.6
Discussion of Va-or Density Results
Vapor density data for nitromethane was not
readily available from the literature. But some
investigators have formulated equations which relate
the vapor density with the temperature.
These are all
based on the Verschaffelt equation (30)
which has the
general form:
d-
dg = d o
1 -
m
(A.2)
d 1 = liquid density
where;
dg
vapor density
= critical temperature
T
T = temperature at which the density is to
be evaluated.
m, d o
= constants, which differ for different liquids.
Density measurements for a number of liquids have
shown (29) these equations to be fairly accurate for
all temperatures except those close to the critical
temperature.
For the case of nitromethane, the general
Verschaffelt equation has been broken down (31)
log
d
= 5.0
- 1.0)
where Tb is the liquid boiling point in K , and
into:
(A.3)
db is
the vapor density at the boiling point, which is given by:
129
0.0122M
db "
where
M is
Tb
(A.4)
the molecular weight.
Another equation for nitromethane, also derived from the
basic Verschaffelt equation, is given (29) by:
log dg
4.44b
T
- 1.
+ log
0.0124M
Tb
(A.5)
where the symbols have the same meaning as in the previous
equations.
Calculations were made for the vapor density of
nitromethane using both equations (
A.3 and A.5 ). The
results obtained from these two equations agreed very
well with each other.
These results were then used as input for the
polynomial regression program so that a polynomial
type equation could be obtained.
This same procedure was used with the propylene
glycol vapor density data.
These equations for vapor density may be in
error by as much as 5%,
(29),
but they are the only ones
available at this time. Therefore, until more experimental
measurements of vapor densities are made, these equations
are our best approximations.
130
A.7
Physical Properties of Methyl Alcohol
Vapor pressure ( Pv in psla, TOF )
i.)
(32)
Pv = 98.6548 - 0.018347(T) + 0.00741241(T)2
OOC< T( 240 OC
2.) Liquid density ( •
e
in lbm/ft 3 , TF )
. 49.5607 - 1.26317x10 4(T)0aC < T < 240 "C
(32)
1.02098x10"4(T)
3.) Vapor density ( p,in lbm/ft 3 , T'F )
2
(29)
= 0.88826 - 0.0136(T) + 4.7952X10I 5 (T)2
78"F<4 T 4 375 F
4.) Surface tension (Tin ibf/ft , TOF)
S= 1.7826x10"
3
(32)
- 3.4611x10" 6 (T)
O0C < T< 150 0 C
5.) Heat of vaporization ( HFG in Btu/1bm, T*F)
(32)
HFG = 527.091 - 0.1779(T) - 0.0011134(T)2
0OC < T < 240 'C
The graphs for methyl alcohol properties are given
in FIG.A.11 through PIG.A.15 .
1 1
VAPOR PRESSURE
TEIIPERA TURE-
~s
MvET/- YLZ ALCOOL,,
ETHYL ALCOHOL
-
160
/
/
/
/20
',,, o
/
/
60
/
o20
I
0
o
60
80
/00oo
/210
FI/G. A 11
-qo /60o
/8
L/0 UID DE7VS/ T Y
vs 7"E/MPERAIT URE
/VE TH yZ- ALC oH0OL
zETIHYL ALCoHOL
0,?
Q,4
0156
0.3
02
20
o
Goo
so
/00o
i o
T-oc
F/ G. A. 12
l-o
/COo
15o
133
VAPOR DE/VS/TY
kYETHYL
vs TE7MPERA TUR E
ALcoHOL
E7TYL AL co//oL
~2/f
2.0
-Q
0.
0.c
0.2
/80
200
2Z0
2oo
2 60
T
0
F
2z0
30o
320
3o
360
154
SURFA CET TE/VSIO/V
1MET//YZ
FT11/YL
vs TEMPERATU/RE
AL C'//oL
ALCoHoL
2#
22.
U
K
20.
/6.
/o,
8.
I/rr
)u
U
-
oU
8c0
loo
/Z o
/ao
/o
Igo
2oo
1-5
1HEA T of VA POR zA T/O/V
vs TEMPERATTURE/
/VLE T// YL AL COAHOL
ALCohOL
CT//Y'L
3oa
Zoo
/8go
/160
N
NNI.
/Zo
p t"
74.
(0L.,
V
...
/00
21
T oc
0
/qo
150
/SO
2co
A.8
Physical Properties of Ethyl Aloohol
1.) Vapor pressure ( P, in psia, ToC)
(32)
2
P, = 38.862 -2.02655(T) + 0.018601(T)
0 C < T< 240 OC
2.) Liquid density (Pein gm/co , TOC )
, 0.8022 - 5.4322x10 4(T) - 3.3149x10-6(T)
(32)
2
0 0C < T< 240 0 C
3.) Vapor density ( 3 in gm/co , T0 C )
(29)
. 8.367x10-3 -3.1762x10"4(T) + 2.541x10" 6 (T)2
68 OF< T<3800 F
4.) Surface tension (T-in dynes/om , TOC )
(32)
- . 224.5354 - 0.09173(T)
00C < T< 150 0C
5.) Heat of Vaporizxation ( HPG in cal/gm , TOC ) (32)
HFG .. 223.88 - 0.13821(T) - 2.0431x10"13(T) 2
OOC<T( 2400 C
The graphs for ethyl alcohol properties are given
in FIG.A.11 through FIG.A.15 *
137
Appendix B
Sample Calculation
Appendix B will give a detailed description of
all calculations necessary to calculate the superheat
temperature,
beginning with the basic equations and
concluding with the required inputs for the computer
program which performs the actual calculations. The sample
calculations will be shown for the case of propylene
glycol.
The starting point for all calculations is
the energy deposition equation described in chapter 2.
This equation is given again below:
/LoE
qe_
A/ dS
2
z)
zR,
-4ý
9 (z,
Yre
.9-
J'4__
\). __
All of the terms in equation (B.1)
have been described
in chapter 2, and will not be repeated here.
The above equation will first be broken down as
far as possible without introducing any specific liquid
properties.
I
138
The second term on the RHS of equation (B.1)
can be
neglected for all of the cases considered in this work.
( The reasons have already been outlined in chapter 2 )
This leaves only one expression on the RHS. The
substitution is
n/
now made for
V
k
p'3
' VWrC'
( Z1 )eff :
V
(B.2)
Cancelling all common terms
-<_c
/V ds
2
2
j
(0Zj)j34
0,'L
(3.3)
-3
where v has been replaced by the energy equivalent,
from the basic equation
E
, MV2
(B.4)
02
The RHS of equation (B.3) is multiplied by
c2 ,(or 1 )
resulting in
i dE
3
-/h1. 0?(BZ5)
This form gives the constants (such as h,c, mo ) in
familiar forms, the values of which can be found in any
nuclear physics text. More specifically, we have:
he = 1.9732 x 10
11
Mev-cm
mo c2 = 0.511 Mev.
mo = 0.5488 x 10- 3
amu.
139
The N term is
replaced by
v MI9EolIl
and brought to the RHS giving
d9-
_
e(A/v
Imv!
015
YF(Z,)
10
(732x/o0
0. X11
d
This is as far as the original equation,
(B.1),
can be
broken down without introducing any specific particle
or medium. The ease of fission fragments in propylene
glycol will now be taken as an example.
The fission fragment properties are as follows
( (5) pg 115 ):
A = 97 = M1
Z = 38 = ZI
The molecular weight of the medium is
Mmedium = Mpropylene
- 76.096
glycol
Substituting the values for 14
M , Z1 , Mmedium , and
for
ds
Nav
(
=
6.023 x 10 - 23, equation (B.7) becomes:
. o)
3.777x/o
)
(.8)
B8
14.0
The summation term is over the atoms in the stopping
medium ( propylene glycol in this case ). The chemical
formula for propylene glycol is
C3H802 , so the
summation sign can be shown as follows: Z
The values for
, and IZ
o
are:
3
Zc = 6
Ic = 76.4ev
S= 8
ZH = 1
I,
Zo = 8
I = 100.0 ev
c
xo
Z-
=
= 2
= 15.5 ev
Values for the mean ionization potential, I- , have
been obtained from Evans (14).
The values for I.,are given in ev, but the 1z term
in equation (B.8) must be in units of Mev ( E is given
in Mev).
Equation (B.8) thus becomes:
7)(B3.9)
141
Equation (B.9) is now reduced to an equation with a
single "In" term:
.01F
Re
3. S7 x /OV(.
04, o
(B.10)
Equation (B.10) is the final form of the energy deposition
equation.
dE/dS is in terms of
Pe
Mev/om
in terms of gm/cc
E in terms of Mev.
The remainder of the calculations provide input
parameters to the computer program.
Equation (B.10) is
substituted into the integral
equation described in chapter 2,
CS.,)
c.(s, )
Performing the integration:
Cs,)
(-.,7 x7z,
/s
ds
l) 1
142
Further breakdown of (8.12) and combination of terms
leads to:
Eu
)-)A
k-(s
The RHS of equation (B.13) is in metric units and is now
converted to allow
Qt
and P* to be input in English units.
The final conversion factor is 2.0479, and the RHS must
be divided by this. (The LHS is in Mev terms and must
remain the same numerical value, no matter what units
the RHS has.)
Converting to English units and substituting the
rela tionship,
2•T
(B.14)
the RHS of equation (B.13) becomes:
( 3 . • Rf
x /O) i
a
(B.15)
- •
The expression in (B.15) is one of the parameters which
is used as an input to the computer program. This parameter
is given the name of "COEF".
COEF =
x3.
ox/o) a
-
(B.1.6)
COEF is
now the RHS of equation (B.13)
, which is
written as
(arrangin)
terms n equation (B.7)
Rearranging terms in
E[(,.) - CO•..E(
(B.17)
SCoEF
equation (B.17)
J,, (s- 3.P'/]
E3.•••4f.-(s,)
3. ,.
-.COL-9J
- E(s).'Coe/F -
£(s,.)
(B.18)
The term enclosed in brackets on the RHS is another
input to the program, and is labeled "DDD" .
DDD =
[2 E (s,)
3.
(B.19)
/! COEFj
The LHS of equation (B.18) can be rewritten in a slightly
different form, and is also used as an input to the
program. The LHS is labeled "CCC" .
ccc
E(s)-COF .•CE(s
,)/
o.o1ro•'~s,)
00 - /
1
(B.20)
1441
Equation (B.18) can thus be written as
EC 4FDZD - COE
006E
CCC
E
ij -
()s
(B.21
The term inside of the brackets is also an input, and
is labeled "BBB"
BBB -
.
DDD - coeF.4F E(s)
(B.22)
There are thus four inputs to the computer program
which were derived from the original energy deposition
formula.
These were:
1.) COEF
2.) CCC
3.) DDD
4.)
BBB
These terms will be different for different fluids.
For the ease of fast neutrons in the medium, the only
difference is that the subscript I in equation (B.2) refers
to the PKOA. In this case there are actually three possible
PKOAs,
namely H , C, and 0. But, as has been shown by
Bell (5), and by El-Nagdy(4)(who went through minimum
superheat calculations for each type of atom),
the
minimum superheat will be obtained with the O-PKOA.
Therefore,
only the 0 (Oxygen) PKOA is
considered.
145
There are some more computer inputs which should
be discussed. ( The computer programs are listed in
Appendix C.).
This is
One input is
the term labeled "EPKA".
the initial energy of the radiation particle,
which has been carried in the equations as
E(si
)
. The
value for E(sl) will always be known.
The physical properties as a function of temperature
are also inputs. These have been given in Appendix A .
There are two final inputs which must be considered.
These are computer statement number
0558 (IF(ABS(E-EC)-X.XXX)9858,0658,0658
and the statement
E = Y.YYYY
The important point here is the value of X.XXX in
statement 0558. This value sets the convergence criteria.
If this convergence criteria is too strict, a COMPLETION
CODE error will result. If
this is
the case, then the
convergence criteria must be relaxed (X.XXX must be made
larger).
The best method of determining what the value of
X.XXX should be, is
with E(s
i
to choose values for E (beginning
) and gradually decreasing in energy) and
actually calculating the value of EC.
EC is
calculated in statement number 0458:
0458 EC = E -(E*E-BBB*E+CCC)/(2.0*E-.BBB)
146
A simple computer program can be written to calculate
EC values for various E values. This will then give an
indication of the minimum value for the ABS(E-EC) term
used in statement 0558. This minimum value is
then used
as the value for X.XXX . ( For the present work the
value of X.XXX varied from 0.0001 to 0.003 )
The value chosen for the E term in
E = Y.YYY
is
not too critical, and an approximate value which can
be used is
:
E =
E(s1
)
,
These ( from COEF to E ) are the factors in the
computer program which must be changed for the different
fluids. Otherwise, the same basic program may be used for
all cases.
147
Appendix C
Computer Programs
This Appendix contains the three basic computer
programs used in this work.
The first one given is the Polynomial Regression
Program which gives us the equations for the physical
properties as a function of temperature. ( Nitromethane is
shown as an example.)
The second program listed in this Appendix is the
program used to calculate the superheat threshold for
the case of fission fragments. ( Benzene parameters are
used as an example.)
The third program is for calculating the superheat
threshold for fast neutrons. ( Benzene parameters are
used as an example.)
C
C.
POLY0001
c
C
$
0lS:41~$t ~-~S4~~~$·t
t
POLYOOO2
POLYO003
POLYO004
POLYO005
POLY0006
~ $SS~SStS~:$S~kS$~r$$~$
POLYNOGFIAL REGRESSION PROGRAM
C
POLY0007
4
NITROMETHANE PROPERTIES
--
SURFACE TENSIC-N
VAPOR PRESSURE
HEAT OF VAPO.IZATION
LIQUID DENSITY
VAPOR DENSITtY
ENGLISH UNITS
LBF/FT
PSIA
a
DEG F
CEG F
a
BTU/LEM
LBM/FT3
LBM/FT3
a
a
CEG F
DEG F
a
DEG F
SAMPLE MAIN PROGRA?' FOR POLYNOMIAL REGRESSION -
POLRG
PURPO SE
(1) READ THE PRCBLEM PARAMETER CARD FOR A POLYNOMIAL REGRESSION, (2) CALL SUBROUTINES TO PERFCRM THE ANALYSIS, (3)
PRINT THE REGRESSION COEFFICIENTS AND ANALYSIS OF VAR.INCE
TABLE FOR POLYAC.IALS OF SUCCESSIVELY INCREASING DEGREES,
POLYOQ08
POLY0009
POLYO0.10
POLY0011
POLY0012
POLYO:13
POLY0014
POLYO015
POLY0016
POLY0017
POLY0018
POLYOO19
POLY0020
POLYOO21
POLYO022
POLY0023
POLY0024
POLY0025
POLY0026
POLY0027
POLY0028
POLY0029
POLY0030
POLY0031
POLYO032
POLY0033
POLY0034
POLY0035
POLY0036
H
AND (4) OPTIONtLLY PRINT TfE TABLE CF RESIDUALS AN[ A PLOT
OF Y VtLUES ANE Y ESTIMATES.
REMARKS
THE NUPBER OF CESERVATICNS, N, MUST BE GREATER THAN M+19
WHERE M IS THE FIGHEST DEGREE POLYNCMIAL SPECIFIED.
IF THERE IS NC FEOUCTICN IN THE RESIDUAL SUM OF SQUARES
BETWEEN TWO SUCCESSIVE DEGREES OF THE PCLYNCNIALS, THE
PRCGRAM TERMINATES THE PROBLEM BEFORE COMPLETING THE AAALYSIS FOR THE HIGCEST DEGREE POLYNOMIAL SFECIFIEC.
SUBROLTINES AND FL.CTION SUBPROGRAMS REQUIRED
GDATA
ORCER
MINVv
MULTR
PLCT
(A SPECIAL PLOT SUBRCUTINE PROVIDED FCOR THE
FROGRAM.)
METHOD
REFER TO B. OSILE, 'STATISTICS IN
COLLEGE PRESS', 1954, CHAPTER 6.
4**
* * * * ***
* *@ *4*,1
*+***1***
+*
1*1*
*
O*
*.*.•4*
*4*
PESEARCH',
, •I• *
*
THE
*
SAMPLE
ICWA STATE
*
5****
THE FOLLCWING DIMENSI(N MUST BE GREATER ThAN CR ECUAL TO THE
FROCUCT OF N*(M+1),
hFERE N IS THE NUMBER OF OeSERVATIONS AND P
IS THE HIGHEST DEGREE POLYNOMIAL SPECIFIED..
GIMENS
DIMENSION X(1100)
THE FOLLOWING DIMENSICN MUST BE GREATER
FRCODUCT CF M*M..
DIMENSION
C
HIAN CR ECLAL TO THE
DI(100)
THE FOLLOWING DIMENSION MUST
BE GREATER THAN CR ECLAL TO
*5
POLYC037
POLY0038
POLY0039
POLY0040
POLY0041
POLY0042
POLY043
POLYO044
POLYO045
POLY0046
POLY0047
POLY0048
POLY0049
POLYO050
POLY0051
POLY0052
POLYO053
POLY0054
POLY0055
POLYO056
POLY0057
POLY0058
POLYO059
POLY0060
POLYOD61
POLY0062
POLY0063
POLYO064
POLY0065
POLY066
POLY0067
POLY0068
POLY0069
POLY0070
POLY0071
POLY0072
9010AtOd
LOTOA1Od
0D10A10d
~0TOA10dd
LOIOTlOd
SIHL HAIM NOI1\fnrNJ33 NI Q&9n S3NI.lO•t 8f3HIO NI ENI8~V3dd
SIN3W31VIS 4'ISI33•ad
31Mf30 W3bAl 39AW3t
303]'13•SN~V1•'8~''
00TOA10d
6600Al0d
6OODAlOd
L6OOAlOd
9600A10d
I o
I'Sb
'
S4BIS
•
I
3H1 '0E3~IS33 SI
"SMD1l3A HIHM LN3~3IViS
WJUd 03ADW3d 3
O011OHS T NOlD03 NI 3
3NIifnOd SIHI 40 NOIS~*A N*DI338d*d 30lfInD V jI
(OOE)d NOISN3WIa
1 01 133 38 AVW4NOISS'43W3 A3
lS 3H41 '39SI483HiL3
E*N 31 lVnb3 bO NVHI
'
I31V3)
398 ISnW '3SV)
SIHi NI 'NDISN3WIG 341 AD 3ZIS 3Hi *038IS30 SI S31VWIIS3 ONV
fIVO 03AB3S83 A3 10Td 3HI AI 03S1 39 111M N3ISN3WIO 3NI001104 3HI
(0l)SNV
LSODA10d
1800Al0d
ELOOAlOd
'BOOlOd
"OT
ELOOAIDd
01 1VB03
80 NVAI
V931g93D3
39
ISnJ
V)ISN3tIO
S1 1)3ASI'(Ti)',SfS'(Ti)30a3'(1t)aLS(II)b 9X
"61t+)
7LODAIOd
3 RHI
NOISID38d 312133 93H
L900A1Qd
8800A1Od
9L00A10d
1L00Al0d
9LOOAIOd
LLOOAOd
9LOOAlOd
OSIV LSITU
S'JX'X NOISI33ýd 319003
9600AlOd
?00AlOd
1600AlOd
0600klOd
6800A10d
32
31
W3f3
MbO NVHI
' 31V3'd
31
170n3
0e NVHI
9NIM3110
131V3'O9 38 ISnW
SN)ISN3WIQ
3HL
NOISNEIWIG
3o8 ISiW SN3ISN3WIO ENIM1-103
(0TL'(01)SS'(0[)3•' (0T)
**W
NOISN'3W10
3HI
NOISN3WIO
NIM37110A
3HL
(99)0 NOI&N3WIf
** /(t+W)*(Z÷+W)
1 FORMAT(A4,A2,15, 12, I)
2 FORM!AT(2F10.0)
3
4
5
6
7
8
FORMAT(27H1PCLYNOMIAL REGRESSICN.....,A4,A2/)
FORMAT(23HONLMBER OF CBSERVATIONS,16//)
FORPAT(32HOPCLYNCMIAL REGRESSION OF DEGREE,13)
FORMAT(12HO
INTERCEF1,E20.7)
FODRMT(26HO
REGRESSICN COEFFICIENTS/(6E20.7))
FOPMAT(1HO/24X,24HANALYSIS OF VAPIANCE FCR,I4,1I9H
DEGREE POLYNODI
IAL/)
9 FOPMAT(lH-0,5X,19HSOUFCE OF VARIATION,7X,9HDEGREE OF,7X,6HSUM OF,tX
1,4HMEAN,IOX,IHF,9X,2CFIMPROVEMENT IN TERMS/33X,7HFREEDOM,8X,7HSQUA
2RES,7X,6HSQUARE,7X•,5•ALUE,8X,17HOF
SUM CF SQUARES)
10 FORMPT(20HO
DUE TO REGRESSION,12X,I6,F17.5, FI4.5,Fl3.5,F2o0.5)
11 FORMAT(32H
DEVIATIIh ABOUT REGRESSICN
,I6,F17.5,F14.5)
12 FORMAT (8X,5HTOTAL,9,I6 ,F17.5// /)
13 FORMAT(17iO
NO IMPRCýEMENT)
14 FORMAT(IH0//27X,18HTAELE OF RESICUALS//16H OBSERVAIICN NC.,5X,7HX
IVALUE,7X,7HY VALUE,7X,10HY ESTIMATE,7X,8FRESIDUAL/)
15 FORMAT(IHO,3X,I65,l.5,F14.5,FI7.5,F15.5)
**
C
.**
**********g*o*
*
****
****0*
.**0*
**
****
READ PROBLEM PARAMETEF CARD
END=1000
100
READ (5,1) PF,PR1,N,P,NPLGT
PR....PROBLEM NUMEER (MAY BE ALPHAMERIC)
PR1...PROELEM NUMEER (CONTINUED)
N ..... NUMBER OF CE-ERVATIENS
M.....l-IGfEST DEGREE POLYNOMIAL SPECIFIED
NFLOT.CPTION CODE FCR PLOTTING
C IF PLOT I! NOT DESIRED.
1
IF PLOT I! DESIRED.
***
*****
*
*
soa*0
*
**
POLY0109
POLY0110
POCLY0111
POLY0112
POLY0113
POLY0114
POLY0115
POLYO116
POLYO 17
POLY0118
POLYO119
POLY0120
POLYO121
POLY0122
POLY0123
POLY0124
POLY0125
POLY0126
POLY0127
POLY0128
POLY0129
POLY0130
POLY0131
POLY0132
POLY0133
POLY0134
POLY0135
POLY0136
POLY0137
POLY0138
POLY0139
POLYOI40
POLY0O14.
POLY0142
POLY0143
POLY0144
PRINT PROBLEM NLMBER
PND N.
'RIITE (6,3) FR,PR1
ýRIT.E. 46,4) AN
READ:INPUT DATA
L=N*
CO 110 I=1,N,
J=L-+I
X(I) IS THE INDEPENDENT VARIAELE, AND X(J)
VtRIABLE.
110 REAC
(5,2)
IS
THE DEPENDENT
X(1),X(J)
CALL GOATA (IMXXBAFSTDDSUMSQ)
tM=M+1
SUM=0.0
NT=N-1
CO 2C0 1=1,M
ISAVE(I)=I
FORM SUBSET
CF CORRELATION COEFFICIENT MATRIX
CALL ORDER (PM,DMM,ItISAVE,DIE)
INVERT THE SLBMATRIX
CF CORRELATION COEFFICIENTS
CALL MINV (DI,ICETET)
CALL MULTR
(N,I,XBAR,STD,SUMSQ,DI,E, ISAVEB,SBT,ANS)
PRINT THE RESULT OF CALCULATION
POLY0145
POLY0146
POLY0147
POLY0148
POLY0149
POLY0150
POLY0151
POLY0152
POLYO153
PGLYO154
POLY0155
POLY0156
POLY0157
POLY0158
PCLYO159
POLY0160
POLY0161
POLY0162
POLY0163
POLY0164
POLY0165
POLY0166
POLY0167
POLYO 168
POLY0169
POLY0170
POLYO0171
POLY0172
POLY0173
POLY0174
POLY0175
POLY0176
POLY0177
POLY0178
POLY0179
POLY0180
kRITE (6,5) I
IF(ANS(7)) 140,130, 130
130 SUYIP=ANS(4)-SUM
IF(SLMIP) 14C, 140, 150
140 WRITE (6,13)
CC TC 210(
150 WRITE (6,6) ANS(L)
%RITE (6,7) (B(J),rJ=rI)
,RITE (6,8) I
hRITE (6,9)
SUP=ANS(4)
ýRITE (6,10) IANS(4),ANS(6),ANS(10),SUMIP
rI=ANS(8)
ARITE (6,11) NI,ANS(7),ANS(9)
WRITE (6,12) NT,SUMSC(MM)
C
SAVE COEFFICIENTS FCF CALCULATION OF Y ESTIMATES
COE (1)=AS( 1 )
CO 160 J=1,I
160 COE(J+1)=E(J)
LA=I
200 CONTINUE
TEST WHETHER PLOT IS
210
IF(NFLCT)
10CC,
100,
CESIRED
220
CALCULATE ESTIMATES
220 NP3=N+N
CO 230 I=I,N
NP3=NP3+1
F(NP3)=CCE(1)
POLY0181
POLY0182
POLYO183
POLY0184
POLY0185
POLY0186
POLY0187
POLYO188
POLY0189
POLY0190
POLY0191
POLY0192
POLY01S3
POLY0194
POLY0195
POLY0196
POLY0197
POLYCI98
POLY0199
POLY0200
POLY0201
POLY0202
POLYO203
POLY0204
POLY0205
POLYO206
POLY0207
POLY0208
POLY0209
POLYO210
POLYO211
POLY0212
POLY0213
POLY0214
POLY0215
POLY0216
F-
230
L=I
CO 230 J=1,LA
F(NP3)=P(KP3)+X(L)*CCE(J4+1)
L=L+h
CCPY CBSEFVED DAT.t
N2=N
L=N*fV
CO 240 I=1,N
F(I)=X(I)
2= N 2+ 1
L=L+1
240 F(N2)=X(L)
PRINT TAELE CF RESIOUILS
250
bRITE (6,3) PR,PR1
%RITE (6,5) LA
bRITE (6,14)
NP2=N
hP3=Nh+N
00 250 I=1,N
hP2=NP2+1
NP3=NP3+i
PESIC=P(NP2 )-P(NP3)
hRITE (6,15) I,P(I),P(NP2),P(NP3),RESID
CALL PLOT (LAP,N,3,C,1)
GO TC 100
1(00
C
STOP
CONTINUE
CALL EXIT
END
POLY0217
POLY0218
POLY0219
POLY0220
POLY0221
POLY0222
POLY0223
POLY0224
POLYO225
POLY0226
POLY0227
POLY0228
POLY0229
POLY0230
POLY0231
POLY0232
POLY0233
POLY0234
POLY0235
POLY0236
POLYO237
POLY0238
POLY0239
POLY0240
POLYO241
POLYO242
POLY0243
POLY0244
POLY0245
POLY0246
POLYO247
POLY0248
POLY0249
POLY0250
POLY0251
POLY0252
\Jn
0s
.4.
00
0a
400
#...4464060.1 0
0
ls
e0.
004
o....
0
00
.............
&
*
.09
s
SUBROUTINE PLOT
PURPOSE
PLOT SEVERAL CRCSS-VARIABLES VERSUS A BASE VARIABLE
L'AGE
CALL PLOT (NO,A,tNMNLNS)
DESCRIPTICN OF PAR/IMETERS
NO - Cl-ART NUMBER (3 DIGITS MAXIMUM)
A - MATRIX OF CATA TO BE PLOTTED. FIRST COLUMN REPRESENTS
BASE VARIAELE AND SUCCESSIVE COLUMNS ARE THE CROSSVARIABLES (MAXIMUM IS 9),
N
- NUMBER OF
FCWS IN MATRIX A
M
- NUMBER OF COLUMNS IN MATRIX A (EQUAL TO THE TCTAL
NUMBER OF VARIABLES). MAXIMUM IS 10.
NL - NUMBER OF LINES IN THE PLCT. IF 0 IS SPECIFIEEt 50
LINES ARE LSED.
NS - CCOE FOR SCRTING THE EASE VARIABLE CATA IN ASCENDING
ORDER
0 SORTING IS NOT NECESSARY (ALREADY IN ASCENDING
CRDE F).
1 SORTING IS NECESSARY.
REMARKS
NONE
SUBROUTINES AND FLNCTION SUBPROGRAMS REQUIRED
NONE
400
.O4000400
0
"64400•
0044r040r
644.0.004000
SUBROUTINE PLCOT(NOtAMNLsNS)
CIEhS ION OUIT(O11), YFP(11), ANG(9 ),A( 1)
040
0404
•...........
000 .
040
.
POLY0253
POLY0254
POLY0255
POLY0256
POLY0257
POLY0258
POLY0259
POLY0260
POLY0261
POLY0262
POLY0263
POLY0264
POLY0265
POLY0266
POLY0267
POLY0268
POLYO269
POLY0270
POLY027i
POLY0272
POLY0273
POLY0274
POLYO275
POLYO276
POLY0277
POLY0278
POLY0279
POLY0280
POLYO28I
POLY0282
POLY0283
POLY0284
POLY0285
POLY0286
POLY0287
POLY0288
CATA
CATA
CATA
CATA
CATA
CATA
CATA
CATA
CATA
CATA
PLANK/' '/
ANG(1)/'1'/
ANG(2)/2'/
ANG(3)/'3'/
ANG(4/1'4'/
ANG(5)/'5'/
ANG(6)/'6'/
ANG(7)/'7'/
ANG(8)/'8'/
ANG(9)/'9'/
C/•T•
EATA
ANG(2|I'2'!
ANG(7|!'7'/
¢,aT• ANG(£;)
CAT•
EATA
CATA
AI•G
ANG(•|
ANG(5)/'5'/
ANG(8)/'8'/
•LAF,(1)
A•G(3}
KI' !'3'/
/'J.''/ /
/14'/
!'9'1
CATA
ANG(6)/'6'!
FOPMAT(1H1,6OX,7H
CV-FT ,13,//)
FORMAT( IH ,F11.4,5X, 1 AI)
FORMtT(F )
FORMAT(1OH 123456789)
FORMAT ( 1OA1)
FCPMAT(1H ,16X,10H.
1
*
*
*
*
.)
8 FOPMtT(1HO,9),11F10. AJ
\LL=I\L
IF(IS)
SORT
16,
16,
10
EASE VARIABLE DATA IN ASCENDING
10 CO 15 I=1,N
CO 14 J=I,N
IF(A(I)-A(J))
11 L=I-N
LL=J-N
CO 12 K=1,M
L=L+I
LL=LL+N
14, 14,
11
CRDER
POLY0289
POLY0290
POLY0291
POLYO292
POLY0293
POLY0294
POLY0295
POLY0296
POLY0297
POLY0298
POLY0299
POLY0300
POLY0301
POLY0302
POLY0303
POLY0304
POLYO305
POLYO306
POLY0307
POLY0308
POLYO309
POLYO310
POLYO311
POLYO312
POLY0313
POLY0314
POLY0315
POLYO316
POLY0317
POLY0318
POLY0319
POLY0320
POLY0321
POLY0322
POLY0323
POLY0324
I
F=A(L)
A(L )=A(LL)
A(LL)=F
CONTINUE
CONTINUE
TEST NLL
16 IFINLL)
18 NLL=.50
20,
18, 20
PRINT TITLE
20 kRITE(6,1)NCO
DEVELCF BLANK ANC EIGITS FOR PRINTING
FINC SCALE FOR BASE VARIABLE
XSCAL=(A(NI-A(11))/(FLCAT(NLL-1))
FINC SCALE FOR CRCSS-VARIABLES
26
28
30
40
t•1=N+.l
VMIN=A(MI
'vAX=YtI I
12=M *N
CO 40 J=M1tM2
IF(A(J)-YNIN) 28,26,26
IF(A(J)-YtMAX) 40,40,{-C
YMIN=A(J)
GO TC 40
YMAX=A(J)
CONTINUE
YSCAL=(YPMAX-lMIN)/1OC.0
POLY0325
POLY0326
POLY0327
POLY0328
POLY0329
POLY0330
POLY0331
POLYO33Z
POLY0333
POLY0334
POLY0335
POLY0336
POLY0337
POLY0338
POLY0339
POLY0340
POLY0341
POLY0342
POLY0343
POLY0344
POLY0345
POLY0346
POLY0347
POLY0348
POLYO.349
POLY0350
POLYO351
POLY0352
POLY0353
POLY0354
POLY0355
POLY0356
POLY0357
POLY0358
POLY0359
POLY0360
C
FIND
BASE VARIABLE PRINT PCSITION
XB=A (1)
L=l
MY=M-I
I=1
45 F=I-1
XPR=XB+F*XSCAL
IF(A(L)-XPR) 50,50,7{
FIND CROSS-VARIABLES
50 CO 55 IX=1,IC1
55 CUT(IX)=BLANK
CO 6C J=1,MY
LL=L J *N
JP=((A(LL)-YFIN)/YSCtL)+1.0
CUT(JP)=ANG( J)
60 CONTINUE
PRINT LINE AND CLEAR,
OR SKIP
hRITE(6,2)XPF,(CUT(IL),IZ=1,i01)
L=L+1
GO TC 80
70 WRITE(6,3)
80 I=I+1
IF(I-NLL) 45, 84, 86
84 XPR=A(N)
GC TC 50
PRINT
CROSS-VARIABLES
86 kRITE(6,7)
YPR(1)=YMIN
NUMBERS
POLY0361
POLY0362
POLY0363
POLY0364
POLY0365
POLY0366
POLY0367
POLY0368
POLY0369
POLY0370
POLY0371
POLY0372
POLY0373
POLY0374
POLY0375
POLY0376
POLYO377
POLY0378
POLY0379
POLY0380
POLYO381
POLY0382
POLY0383
POLY0384
POLY0385
POLY0386
POLYO387
POLY0388
POLY0389
POLY0390
POLY0391
POLYO392
POLY0393
POLY0394
POLY0395
POLY0396
CO 90 KN=1,9
90 YPR(KN+1)=YPP(KNI+YSCAL*10.0
YPR(11)=YNAX
kRITE(6,81(YFR(IP),IF=I,1 )
RETURN
END
POLY0397
POLY0398
POLY0399
POLY0400
POLYO401
POLYO402
C
C
C
C
*********************
C
*
C
C
C
C
C
*
******************
*******************
FISSICN FRAGMENTS IN BENZENE
4
*
ENERGY BALANCE METHED
A
*
C
C
C
C
C
IMPLICIT REAL*8
EXP( X)=DEXP( X)
(A-H,{-.Z)
ALCG(X)=CLOG(X)
SQRT (X)=DSQR T(X)
ABS( X)=DABS ( ý)
CIMENSION PLIQ(30),TSAT(30)
CIMENS ION A(30)
500 FORMAT(F10.4)
501 FOPMAT(I3/(10X,2F10.2))
THE SUPERHEAT LIMIT FOR
503 FORMAT(1H1,30X,'
I TO FISSICN FRAGMENTS '////)
ALPHA
A PKC GT EPKA DELTI
504 FORMAT(lHO,119H•
GAS
VAPCR
XNI
REL SIZE
J
IVG ENER
EXPOSEC
mNZENE
PRESS
DEL T
SAT
DEL
T
ENER
A
SUP
FFBZ0001
FFBZ0002
FFBZ0003
FFBZ0004
FF810005
FFBZ0006
FFBZ0007
FFBZO008
FF810009
FFBZ0010
FF8Z0011
FFBZ0012
FFBZ0013
FFBZ10014
FFBZ0015
FFBZ0016
FFBZ0017
FFBZOOI7
FFBZ0018
FFBZ0019
FFZO0020
FF8Z0021
FFBZ0022
FFB10023
FFBZ0024
FFBZ0025
FFBZ0026
FFBZ0027
FFBZ0028
FFBZ0029
FFBZ0030
FFBZ0031
FFBZ0032
FFBZ0033
FFBZ0034
FF81Z035
FFBZ0036
p
O)
i
1 )
LIQUID
TEMP
LEVEL C
RATE
TIME
505 FORMATIlH 119gH
PRESS
EQA48
E.B.M.
TEMP
INIT EMBR
PRESS
1EP RATE
1 )
MEV fV
LBF/FT2 DEG F
SEC
NC/SEC
506 FORMAT(1hO,119H
DEG F
LBF/FT2 LBF/FT2
CEG F
DEG F
1EV/CM
1
,
508 FORMAT(IH ,F5.2,D6.3 ,08.1,F6.3,F8.1,F8.3,F9.5,O1C.3, I5,F5.v1F10.8
It
F7.
Ft E.s1
EXtF8.3
FE.3)
600 FOPMAT(1H0,2SHTHE VALLE OF THE CONSTANT IS ,F1C.4//1
S99 FO.PAT(I3/(F10.3))
1.4///1
01. FORMAT(8H EPKA = tF
C
C
EPKA IS THE INITIAL ENERGY CF TI-E FISSICN FRAGMENT
1: REAt (5.501)N. (PLQ(Lt) ,TSATi
Ml
=,,K
REAC (: i.999) K, (A
3 WRITE(6,
:03)
4 WRITE(6t.504)
5 WRI TE(6,505)
)
L= ,lN)
6 WRI(TE(; 606)
000 READc(5:,500)0CCNST
WRiTE( 6, 600) CGNST
R ITE(6, 601 )EPKA
'cO 110 M=1, K,1
8 CO 109 L= ,N,•
10o tEL F=10 .0
0C
C
FF810043
FF1Z0044
FFBZ0045
FFBZ0046
FF8Z0047
FFZ00o48
FF8Z0049
FFBZ 0050
FFBZ0051
FFBZ00o 52
FF8t0053
EPKA = 95.0
ci
FFZ81037
FFBZ0038
FFBZ0039
FFBZ0040
FFBZ0041
FFB10042
15 f1f:SAT-(L)+OELTF
23 CONTINUE:
PHYSICAL PROPERTIES OF BENZENE
C
25 PVA = 85.14.3 - 0 .959.4TF + 0.003178t*TF*TF
26 PVPSF = PVA*I.44.0
FFBZ0054
FFBZ0055
FF810056
FF810057
FFBO058
FFBZ0059
FFB61060
FFB810061
FFBZ0062
FF8 Z063
FFZ82064
FF 8Z0065
FFB70066
FF8Z0067
FFZ810068
FFOt0069
FFBZ0070
FFBZ0071
FFBZ1002
0H
1
E
SIGMA = 2.2tEE-3 - 4.4l1CE-6*TF
PCWL =
. C/(C .1831
4 6.15524E-5*TF + 0.3444E-7*TF*TF)
ROWV = 1.C/(26.E282 - C.1614*TF + 0.2487EE-3*TF*TF)
FFC = 196.1ii
- U.12169*TF - 0.1734E-3*TF*TF
eA=(PVPSF-PL IC(L) /PýFSF
E = 10.0
RPJWL=CIN ST*RCWL
eENZENE
PARAMETERS
FOR
FISSION
FRAGMENTS
CI53
C259
3258
C3?5
C)EF = (3.4212E04)*RCtL*A(M)*SIGtA/(BA*PlPSF)
CCC = EPKA*EFKA - CCEF*EPKA•(ALC-(O.0622,iEPKAI-1.0)
EDC = 2.C4EPKA + 3.777*CCEF
EeR = CCC - CCEF*ALCG(E)
(458
0O558
EC=E-( E*E-BrE *E+CCC)/(2.0*E-8B8)
IF((A BS(E-EC)-0.0JIC).LT.O.0)
GO TO
F = ( E + EC )* 0.5
IF ( F.LE.0.Cl
E = I.C
9858
C758 GC TC 0358
Ef5~
62
IF((EC-1C.OC).LT.O.L) GC TO 96
IF((FC-EPKAA).GE.0.OI C-C TO 93
CEDCS = (EPKA-E)*BA*PVFSF/(60.96*4(M)*SIGtA)
RCPIT=2.C*SICMA/(BA*PPSF)
CATE=3.7GO221E-19*A(-t )*(TF+459.b91)*DEDS/SIGMA**2
ART=(2.*i;GATE*(PVPSF-FLIQ(L) )-I.U)**2-4.C*GATE**2*(PVPSF-PLIQ(L)
IF(RT.LT.. .C)
GC TC 'S3
,3 PGPSF =( 1.-2.C*GATE*(PVPSF -PLIC(L))-SF'FT(AHT))/(2.0*GAIE)
64 E=(PVPSF +PCFSF -PLIC(L))/PVPSF
)*
FFBZ0073
FF8Z0074
FFBZ0075
FFBZ0076
FFBZC077
FFPZ0078
FFBZ0079
FFBZO080
FFBZ0081
FFBZ0082
FFBZ0083
FFBZ0084
FFBZ0085
FFBZ0086
FFBZ0087
FFBZ0088
FFBZ0089
FFBZ0090
FFBZ0091
FFBZ0092
FFBZOOS3
FFBZ0094
FFBZ0095
FFBZ0096
FFBZ0097
FFBZO0S8
FFBZ0099
FFBZ10100
FFBZCI01
FFBZ0102
FFBZ0103
FFBZ0104
FFBZ0105
FFBZ106l
FFBZ0107
FFBZ0108
C164
IF((8-BA).LT.O.00001)
GO TO 464
BA = B
0364 CO TC 0158
464 GO TC 103
V*HFG*778.26-RCRIT*SIGP'A
103 EQA53=0.8010E-12*A(Mý *DEDS-RCRI T*RCRIT*RCV
104 IF(ECA53-0.0)9696,106,9393
9696 IF(A8S(ECA53).LE.I.0E-16) GO TO 106
GO TO 96
9393 IF(APS(ECA53).LE.1.0E-16) GO TO 106
CO TO 93
93 TF=TF-DELTF
94 CELTF=0.1*DELTF
95 IF(DELTF.LT.0.009) GO TO 106
96 TF=TF+DELTF
97 GO TC 23
106 CELT3=TF-TSATI(L
FKC2PT=O.C
XI=0.0O
CELT I=0.0
ALPHA=O.C
J=0
XN1=O.0
107 WRI TE(6,508)A(M) ,PK02FTOELT IALFHAtPL IQ(L),TST(L),E
,CEDS
,J,
1XN I, XI,PGPSF,PVPSF, DELT3, TF
109 CONTINUE
7 WRITE(6,SC0O)
DIFFERENT VALUE OF A ,//)
900 FORMAT(//45X,221
110 CONTINUE
CO TO 1000
END
FFBZ0109
FF8Z0110
FFBZ0111
FFBZ0112
FFBZ0113
FFBZ0114
FFBZ0115
FFBZ0116
FFBZ0117
FFBZ0118
FF8Z0119
FFBZ0120
FFBZ0121
FFBZ0122
FFBZ0123
FFBZ0124
FF8Z0125
FFBZ0126
FFBL0127
FFBZ0128
FFBZ0129
FFZ80130
FFBZ0131
FFBZ0132
FFBZ0133
FFBZ0134
FFBZ0135
FFBZ0136
FFBZ0137
FFBZ0138
----
L-
N
n
li
FNBZ0001
FNBZ0002
FNBZ0003
FNBZ0004
FNBZ0005
FNBZL006
FNBZ0007
FNBZ0008
FNBZO0009
*
C
FAST
KIEUTRONS IN BENZENE
ENERGY BALANCE
METHOD
FNBZ0010
IMPLICIT REAL*8 (A-H,C-1)
EXP(X)=DEXP(X)
ALOG(X )=DLOG X)
SQRT (X)=DSQRT(X)
ABS (X)=DABS (m )
IIMEhSION PLIQ(30),TSMT(30)
CIMENSION A(30)
500 FORMAT(F10.4)
501 FORMAT(I3/(1OX 2F10. 2))
503 FORMAT(IH1t30X,' THE SUPERHEAT LIMIT FOR
'////)
1 TO FAST NEUTRONS
ALPHA
504 FORMAT(IH0,119H A PKO GT EPKA DELTI
GAS
VAPOR
XN.I
REL SIZE
J
1VG ENER
1
EENZENE
EXPOSEC
ENER A
SUP
PRESS
DEL T
SAT
DEL
LIQUIC
EQA48
LEVEL C
TEMP
E.B.,M,
TEMP
T
)
505 FORMAT(IH ,119H
1EP RATE
1
)
506 FORMAT(1HO 119H
1EV/CM
RATE
INIT EMBR
NC/SEC
TIME
PRESS
PRESS
LBF/FT2 DEG F
SEC
DEG F
LBF/FT2 LBF/FT 2 DEG F
MEV
M
DEG F
FNBZ0011
FtNB10012
FNBZ0013
FNBZ0014
FNBZOOI5
FNBZ0016
FNBZ0017
FNBZOO18
FNBZ0019
FNBZ020
FNBZ0021
FNBZ0022
FNBZ0023
FNBZ0024
FNB0025
FNBZ0026
FNBZ027
FNBZ0028
FNBZ0029
FNBZ0030
FNBZ0031
FNBZ0032
FNBZ0033
FNBZOO034
FNBL0035
FNBZ0036
1 )
508 FORMATI(H ,F5.2,06.3
I
F7 I,-
,D8.1,F6.3,F8.1,F8.3,F9.5,010.3,I5,F5.1,F10.8
)
F7.,F.,XF8.3,F.3
,F8,
600 FORMAT(1HO,29HTHE VALLE OF THE CCNSTANT
999 FOPMAT(13/(F10.3))
601 FORMIAT(8H EPKA = ,FIO.4///)
IS ,FIC.4//)
EPKA IS THE ENERGY OF THE PKOA
C
EPKA = 2.12
C
1 READ(5,501)N,(PLIQ(L),TSAT(L),L=1,N)
READ(5,999)K ,(A(M) ,M=,K
3 WRITE(6,503)
4 WRITE( 6,504)
5 WRITE(6,505)
6 WRITE(6,506)
100 0 READ(5,500)CCNST
WRITE(6,600 )CONST
WRITE(6,60C )EPKA
CO 110 M=1,K,1
8 CO 109 L=1,N,1
10 CELTF=10.C
1 5 TF=TSAT(L)+DELTF
2 3 CONTINUE
PHYSICAL PROPERTIES OF BENZENE
PVA = 85.143 - 0.9594*TF + 0.003178*TF*TF
PVPSF = PVA*144.0
SIGMA = 2.26EE-3 - 4.610E-6*TF
POWL = 1.0/(0.01831 + 0.15524E-5*TF + 0.3444E-7*TF*T*TF)
ROWV = 1.01(26.8282 - 0.1614*TF + 0.24878E-3*TF*TF)
fFG = 196.119 - 0.121t9*TF - 0.1734E-3*TF*TF
BA=(PVPSF-PLIQ(L)) /PVPSF
FNBZ0037
FNBZ0038
FNBZ0039
FNBZO040
FNBZO041
FNBZ0042
FNBZ0043
FNBZ0044
FNBZ0045
FNBZ0046
FNBZ0047
FNBZ0048
FNBZO049
FNBZ0050
FNBZO051
FNBZ0052
FNBZ 0053
FNBZ0054
FNBZ0055
FNBZ0056
FNBZOO57
FNBZ0058
FNBZ0059
FNBZ0060
FNBZ061
FNBZ0062
FNBZ0063
FNBZ0064
FNBZ065
FNBZ0066
FNBZ0067
FNBZ0068
FNBZ0069
FNBZ0070
FNBZ0071
FNBZO072
H
V.I
E =
1.50
ROWL=CONST*RCWL
0158
0258
3258
0358
COEF = (1.00E04*ROWL*A(M)*SIGMA)/(BA*PVPSF)
CCC = EPKA*EPKA - COEF*EPKA*(ALOG(O.9450*EPKA)-1*.)
ODD = 1.0565*COEF + 2.0*EPKA
BBB = 0DD - COEF*ALOG(E)
0458 EC=E-(E*E-BBE*E+CCC)/(2.0*E-BB)
0558 IF((ABS(E-ECJ-0.0030).LT.O.0) GO TO 9858
E = ( E + EC 1~ 0.5
IF ( E.LE.0.0C
E = 1.C
0758
9858
62
63
64
0164
0364
464
103
104
GO TO 0358
IF((EC-1O.00).LT.O.0) GO TO 96
IF((EC-EPKA).GE.0.0) GO TO 93
GEDS = (EPKA-EJ*BA*PVPSF/(60.96*A(M)*SIGMA)
RCRIT=2.O*SIGMAI(BA*PPSFI
GATE=3.700221E-19*A M *(TF+459.69)*DEDS/SIGMA*,*2
ART=(2.0*GATE*(PVPSF-FLIQ(L) )-I.0)**2-4.0*GATE**2*(PVPSF-PLIo(L ) •
1*2
IF(ART.LT.O.0) GO TO S3
PGPSF =(1.0-2.0*GATE*(PVPSF -PLIC(L))-SQFT(ART))/(2.0*GATE)
P=(PVPSF +PGPSF -PLIQ(L))/PVPSF
IF((B-BA).LT.O.00001 ) GO TO 464
EA = 8
CO TO 0158
GO TC 103
EQA53=0. 801E-12*A( M*DEDS-RCRI TRCRIT*RCWV*HFG*778.26-RCRIT*S IGMV
IFIEQA53-0.0)9696,106,9393
9696 IF(ABS(EQA53).LE.i.OE-16)
GO TO 96
9393 IF(ABS(EQA53).LE.1.OE-16)
GO TC 9.3
93 TF=TF-DELTF
GO TO
106
GO TO
106
FNBZ0073
FNBZ0074
FNBZ 0075
FNBZ0076
FNBZ0077
FNBZ0078
FNBZ0079
FNBZ0080
FNBZ0081
FNBZ0082
FNBZO83
FNZ80084
FNBZ0085
FNBLZ086
FNBZ0087
FNBZ0088
FNBZOO89
FNBZO090
FNBZ0091
FNBZ0092
FNBZ 0093
FNBZ0094
FNBZ0095
FNBLOO96
FNBZ0097
FNBZ0098
FNBZ0099
FNBZ0100
FNBZ0101
FNBZ0102
FNBZ0103
FNBZ0104
FNBZO105
FNBZ0106
FNBZ0107
FNBZO108
ON
94 CELTF=0.1*DELTF
95 IF(DELTF.LT.C.009) GC TO 106
96 TF=TF+DELTF
97 GC TC 23
106 CELT3=TF-TSAT(L)
FK02RT=0 .0
XI=0.0
DELTI=0.0
ALPHA=0.0
J=O
XN1=0.0
107 WRITE(6,508 ) (M),PK02T,DELTI,ALPHA,PLIQ(L),TSAT(L),E
IXNI,XI ,PGPSFPVPSFDELT3,TF
109 CONTINUE
7 hRITE(6,900)
900 FORMAT(//45X,22H DIFFERENT VALUE OF A ,//)
110 CONTINUE
GO TC 1000
END
,DEDS
,J,
FNBZ0109
FNBZ0110
FNBZ0111
FNBZOI12
FNBZ0113
FNBZ0114
FNBZ0115
FNBZ0116
FNBO0117
FNBZ0118
FNBZ0119
FNBZ0120
FNBZ0121
FNBZ0122
FNBZ0123
FNBZ0124
FNBZ0125
FNBZ0126
FNBZ0127
168
Appendix
D
References
1.
Glaser, D.A.
2.
Becker, A.G.
"Effect of Nuclear Radiation on the
Superheated Liquid State." Thesis, Wayne State
Phys. Rev
665(1952)
University, Detroit, 1963
3,
Deitrich, L.W. "A Study of Fission-FragmentInduced Nucleation of Rubbles in Superheated Water."
Technical Report No. SU-326-P13-14 Stanford Univ.
1969
4.
M.M. El-Nagdy, "An Experimental Study of the
Influence of Nuclear Radiation on Boiling in
Superheated Liquids."
Thesis, University of
Manchester, England.
5.
"Radiation Induced Nucleation of the
Bell, C.R.
Vapor Phase."
Thesis,
M.I.T., Cambridge, Mass. 1970
6.
Tso, Chih-Ping
"Some Aspects of RadiationInduced Nucleation in Water." Thesis, M.I.T.
1970
7.
Glaser, D.A.
8.
Seitz, F.
9.
Glaser, D.A.
"The Bubble Chamber"
Encyclopedia of Physics Vol XLV S. Flugge ed. (1958)
10.
Peyrou, C. H. "Bubble Chamber Principles"
Bubble and Spark Chambers , R.P.Shutt, ed. 1967)
11.
Norman, A. and Spiegler, P.
Nuclear.Science
And Engineering : 16
213-217 (1963)
Nuovo Cimento
Phys. Fluids
1
II
Supp.2, 361(1954)
2(1958)
~~_li_
169
12.
Deitrich, L.W.
Nuclear Science and Engineering
45
218-232 (1971)
13.
Experimental Nuclear Physics
begre, E.
John Wiley and Sons. (190)
14.
Evans, R.D.
McGraw-Hill
15.
Nuclear Reactor
Glasstone, S. and jesonske, A.
Engineering
D. Van Nostrand Inc.
(1965)
16.
Claxton, K.T. "The Influence of Radiation on the
Inception of "oiling Liquid Sodium"
U.K.A.E.A.,
Harwell, IEngland
17.
Mellan, I.
Compatibility and Solubility
1968
Noys Data Corp.
18.
Timmermans, J.
"Physico-chemical Constants of
Pure Organic Compounds"
Elsevier Pub. Co. (1965)
19.
International Critical Tables
Vol I
The Atomic Nucleus
(0195)
Vol.
III, IV
20. Marsden and Mann
, Solvents Guide
Interscience Publishers, 1963.
21.
CRC Handbook of Chemistry and Physics
22.
Riddick, J.A. and Bunger, W.B.
Wiley-Interscience
1970
23.
Surface Active Agents
Young, C. and Coons, K.W.
Chemical Publishing Co. ,Inc. (1945)
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